quantum and classical algorithms for graph classi cation

Quantum and classical algorithms for graphclassification and search problems

This thesis is presentedfor the degree of Doctor of Philosophy of

The University of Western Australia

Brendan DouglasBSc (Adv. Sc.) (Hons)

School of PhysicsThe University of Western Australia

July 2011

Abstract

This thesis is split into three parts, each addressing the overarching theme of applying

graph theoretic techniques to classification and search problems. Part I explores this theme

in the context of quantum algorithms, in particular involving quantum walks, coined by

Aharonov et al. in 1993 [1]. As a quantum analogue to classical random walks, quan-

tum walks represent a promising candidate for quantum algorithms providing speed-ups

over known classical methods. I will discuss two categories of quantum walk algorithms;

quantum search and graph isomorphism. In particular, the efficiency of quantum walk

based search algorithms on various families of graphs will be investigated, linking both

efficiency of quantum circuit implementation and the complexity of the search problem

to the symmetry of the underlying graphs. Several families of graphs are presented that

are amenable to efficient quantum walk based searching, together with explicit quantum

circuit implementations. In the case of the n-cube, I show that O(√

2n) steps of a quanum

walk are sufficient to find a marked sub-d-cube (where d << n), generalising the results

of Shenvi et al. [2]. The level of structural information provided by walking on a graph

is explored, leading to the construction of a quantum walk graph isomorphism algorithm.

This is shown to efficiently characterise all strongly regular graphs up to order 64 as well

as a range of other graphs tested, including vertex-transitive graphs and projective planes.

Building on this quantum walk based graph isomorphism algorithm, Part II links this al-

gorithm to a well-known classical algorithm, termed the k-dimensional Weisfeiler-Leman

(k-dim WL) method. Following its introduction in the 1970’s, this method was considered

to be a possible candidate for a solution to the GI problem. However subsequent, seminal

papers by Cai, Furer and Immerman [3], and Evdokimov and Ponomarenko [4] seemed

to eliminate this consideration, presenting families of non-isomorphic pairs of graphs that

the WL method cannot distinguish in polynomial time, relative to graph size. Indeed,

following the work of Cai, Furer and Immerman [3], the question of whether the WL

method or some minor variation might solve GI has generally been considered closed.

iii

Properties of these counterexample graphs are investigated, and I show that although a

direct implementation of the k-dim WL method fails to characterise them, under rela-

tively unrestrictive assumptions it does refine their vertex sets to the orbits of the related

automorphism group. Hence I construct an extension to the recursive k-dim WL method,

and prove that it efficiently distinguishes all known counterexample graphs.

In Part III a further implementation of these general graph characterisation methods is

presented, efficiently classifying the automorphism groups of a family of graphs known

as Adinkras, introduced by Faux and Gates [5] to study off-shell representations of su-

persymmetry. A set of local parameters is shown to classify Adinkras according to their

equivalence and isomorphism classes, representing further progress in this recent pursuit

of a solution to the long-standing ‘off-shell’ problem of supersymmetry. Previous results

dealing with characterisation of Adinkra degeneracy via matrix products are extended,

and algorithms are presented for calculating the automorphism groups of Adinkras and

partitioning Adinkras into their isomorphism classes.

iv

Contents

Abstract iii

Acknowledgements ix

List of Publications xi

Preface xiii

1 Introduction 1

I Quantum Walks 5

2 Properties of Quantum Walks 9

2.1 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Algorithmic Applications of Quantum Walks . . . . . . . . . . . . . . . . . 11

2.2.1 Quantum Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Faster Hitting Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.3 Search Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.4 Element Distinctness . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.5 Graph Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.6 Quantum Simulation and Universal Quantum Computers . . . . . . 15

2.2.7 Quantum Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Quantum Walk Based Search Algorithms 19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Metrics of Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Efficient Circuit Implementation . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 Circuit Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Efficient Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.5 Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.1 Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5.2 Hypercube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5.3 Complete Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5.4 Twisted Toroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5.5 Glued Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Search for a Marked Subcube of a Hypercube . . . . . . . . . . . . . . . . . 38

v

3.6.1 Walk on an Unmarked Hypercube . . . . . . . . . . . . . . . . . . . 38

3.6.2 The Marked Hypercube . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6.3 Collapsing to a Walk on the Line . . . . . . . . . . . . . . . . . . . . 40

3.6.4 Search Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.5 Proof of Running Time . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 Conclusions and Open Questions . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Quantum Walk Based Graph Isomorphism Algorithm 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Quantum Walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Proposed GI Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.2 Discussion of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.3 Complexity of Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.4 Addendum to the Algorithm - Finding an Isomorphism . . . . . . . 66

4.4 Testing Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Quantum Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.7 Subsequent Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

II The Weisfeiler-Leman Method and Graph Isomorphism Testing 75

5 Background and Known Counterexamples 79

5.1 Graph Theoretic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 The Graph Isomorphism Problem . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 A General Background . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 The Weisfeiler-Leman method . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Formal Description of WL Method . . . . . . . . . . . . . . . . . . . . . . . 82

5.3.1 The Quantum Walk Phase Method . . . . . . . . . . . . . . . . . . . 85

5.3.2 CFI Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.4 Coherent Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.2 Weak and Strong Isomorphisms . . . . . . . . . . . . . . . . . . . . . 90

5.4.3 Coherent Configurations of Graphs . . . . . . . . . . . . . . . . . . . 91

5.4.4 m-Extended Coherent Configurations . . . . . . . . . . . . . . . . . 92

5.4.5 Examples of Non-Isomorphic k-similar Coherent Configurations . . . 92

5.4.6 The Associated Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 94

vi

6 A Modified Weisfeiler-Leman method 966.1 Graph Extensions and the k-dim WL Method . . . . . . . . . . . . . . . . . 96

6.1.1 Properties of the CFI graph extension . . . . . . . . . . . . . . . . . 97

6.1.2 K(G) Counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.1.3 General graph extensions . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2 Orbit Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.2.1 Counterexamples (orbit case) . . . . . . . . . . . . . . . . . . . . . . 107

6.3 Properties of the WL method . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.4 Extended modular decomposition method . . . . . . . . . . . . . . . . . . . 120

6.4.1 Preliminary definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.2 Graphs under consideration . . . . . . . . . . . . . . . . . . . . . . . 122

6.4.3 Decomposition method . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.4.4 Discussion of Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5.1 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

III Adinkras and Off-Shell Representations of Supersymmetry 135

7 SUSY and Adinkras 1397.1 A Brief Introduction to Supersymmetry . . . . . . . . . . . . . . . . . . . . 139

7.1.1 Garden Algebras and Adinkras . . . . . . . . . . . . . . . . . . . . . 141

7.2 Formal Definition of Adinkras . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.2.1 Graph-Theoretic Notation . . . . . . . . . . . . . . . . . . . . . . . . 147

7.2.2 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . . . 148

7.3 Alternative Representations and Models of Adinkras . . . . . . . . . . . . . 150

7.3.1 Equivalence and Isomorphism Definitions . . . . . . . . . . . . . . . 150

7.3.2 Linear Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527.3.3 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8 Classification of Adinkras 1608.1 Automorphism Group Properties of Adinkras . . . . . . . . . . . . . . . . . 160

8.2 Characterising Adinkra Degeneracy . . . . . . . . . . . . . . . . . . . . . . . 165

8.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1658.2.2 Degeneracy for General N . . . . . . . . . . . . . . . . . . . . . . . . 167

8.3 Identifying Isomorphism Classes of Adinkras . . . . . . . . . . . . . . . . . 170

8.3.1 Partitioning into Isomorphism Classes: Further Examples . . . . . . 174

8.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1788.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.5.1 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9 Conclusion 183

Bibliography 185

vii

Acknowledgements

This ending to my doctoral studies at UWA has been a long time coming, and I am

indebted to many people both for support throughout my time here and for help compiling

this thesis. I am very grateful for the help and support provided by supervisor Prof.

Jingbo Wang, who gave up so much time from her busy schedule to provide direction and

enthusiasm for this work, and without whom this thesis would certainly not have been

completed.

Thank you to Prof. Jim Gates for his time and patience teaching me about Adinkras and

supersymmetry, and for his support and guidance during this work. I would also like to

acknowledge financial assistance from a UWA Hackett Scholarship, and later from a UWA

Completion Scholarship.

UWA has been a great place to be over the years of this study - the friends and fellow

students here have made this time very enjoyable. Thanks go to Terence Peters for reading

through a draft copy of this work and pointing out numerous improvements, and to Tom

Loke for finding and fixing errors in the original versions of the cycle and glued-tree circuit

diagrams included here.

Finally thank you to all my family and friends, for making these years the best of my life

(so far).

ix

List of Publications

This thesis is based in part on the following papers, either published or currently under

submission.

1. B.L. Douglas and J.B. Wang, 2008. A classical approach to the graph isomor-

phism problem using quantum walks, J. Phys. A: Math. Theor. 41, 075303,

arXiv:0705.2531 [quant-ph]. (Chapter 4)

2. B.L. Douglas and J.B. Wang, 2009. Efficient quantum circuit implementation of

quantum walks, Phys. Rev. A. 79, 052335, arXiv:0706.0304 [quant-ph]. (Chapter 3)

3. B.L. Douglas and J.B. Wang, 2009. Erratum: Efficient quantum circuit implemen-

tation of quantum walks, Phys. Rev. A. 80, 059901E. (Chapter 3)

4. B.L. Douglas, S. James Gates, Jr. and J.B. Wang, 2010. Automorphism properties

of Adinkras, arXiv:1009.1449 [hep-th]. Under submission. (Chapter 8)

5. B.L. Douglas, 2011. The Weisfeiler-Leman Method and Graph Isomorphism Testing,

arXiv:1101.5211 [math.CO]. Under submission. (Chapters 5 and 6)

I am the first author of all the publications listed above, of which my contribution is

above 80%, including the drafting and preparation of all manuscripts. Permission has

been granted to include this work.

Brendan Douglas

Jingbo Wang

xi

The following is a list of conference presentations and proceedings arising from this thesis.

1. J.B. Wang and B.L. Douglas, 2007. Quantum walks on graphs, Centre for Quantum

Computing Technology Annual Workshop (February 2007, Sydney, Australia).

2. B.L. Douglas and J.B. Wang, 2007. Efficient Implementation of Quantum Walks,

The Eleventh Workshop on Quantum Information Processing (December 2007, New

Delhi, India).

3. J.B. Wang and B.L. Douglas, 2007. Graph identification by quantum walks, The

Eleventh Workshop on Quantum Information Processing (December 2007, New Delhi,

India).

4. J.B. Wang, K. Manouchehri and B.L. Douglas, 2008. Physical Implementation of

Quantum Walks with Quantum Dots, The 2nd International Workshop on Solid State

Quantum Computing (June 2008, Taipei, Taiwan).

5. J.B. Wang and B.L. Douglas, 2009. Efficient quantum circuit implementation of

quantum walks, The 4th Workshop on Theory of Quantum Computation, Commu-

nication, and Cryptography (May 2009, Waterloo, Canada).

6. J.B. Wang and B.L. Douglas, 2009. Quantum random walk, physical implementation

and potential applications, 10th International Symposium on Frontiers of Fundamen-

tal and Computational Physics (November 2009, Perth).

7. B. L. Douglas, 2010. Automorphism properties of Adinkras, AMSI Summer School

(July 2010, Brisbane, Australia).

xii

Preface

Scientific research in any area is rarely a linear progression, following a distinct path with

clearly defined points of progress. As such, pursuing the answer to a single, initially narrow

question often leads the researcher to a variety of seemingly disparate fields, a process that

will become quite evident in the contents and structure of this thesis.

What initially began as a study of quantum walk algorithms for graph characterisation,

detouring somewhat briefly into quantum walk search algorithms, soon became purely

involved with classical graph isomorphism algorithms. Just when it appeared to have

strayed forever into pure maths†, a set of unexpected discussions led to the realisation that

these same general graph classification methods represented a neat solution to a problem

in a recently developed area of supersymmetric representation theory. This (approximate)

path from physics to maths and back again (broadly speaking), is reflected in the structure

of this thesis, which is split into three parts, covering in turn the work on quantum walks,

classical graph isomorphism methods, and graphical depictions of off-shell representations

of supersymmetry.

Given the range of topics covered, each of the three parts is designed to be in some

sense stand-alone, in that each should be potentially readable in isolation. Although

there are strong links between the topics covered, representing a clear continuation in the

development of the contained ideas, connected by a general focus on graph classification

and isomorphism, it is hoped that the reader can approach this work by reading the parts

in any order, if so desired.

Consequently, each of the three parts contains an introduction to the topic area covered

and the results obtained. As a result, some concepts and definitions are by necessity

repeated, however in these cases I have made an effort to keep them as brief as possible.

†Probably a common occurrence, and not always unwelcome.

xiii

Chapter 1

Introduction

This introduction will contain details of each of the three general subject areas comprising

this thesis, providing a summary of the contributions of each.

The field of quantum complexity, while relatively new, has produced several deep results

that have changed the way we look at computational complexity. Arguably beginning with

Shor’s algorithm for integer factorisation [6], quantum algorithms have demonstrated the

potential for significant speed-ups over their classical counterparts. In cases such as Shor’s

algorithm, these results represent exponential speed-ups over the best known classical

algorithms. In other cases, perhaps most notably Grover’s algorithm for quantum search

of an unsorted database [7], the quantum algorithms, phrased relative to an oracle, allow

speed-ups over the best possible classical algorithms.

Quantum walks represent a particularly fruitful branch of quantum algorithm develop-

ment, and form the focus of the first part of this thesis. As opposed to the typical

form of their classical counterpart in classical random walks, quantum walks are a de-

terministic process, exhibiting often counter-intuitive dynamics and potentially useful

properties, which have already been employed in constructing a variety of quantum algo-

rithms [2, 8–11].

Search problems in particular have represented a rich area for quantum walk algorithms.

Beginning with the work of Shenvi et al. [2], which recast the essential properties of

Grover’s search algorithm in terms of a quantum walk algorithm, quantum walks have

been shown to yield algorithmic speed-ups over the classical case when searching on several

classes of highly symmetric graphs [9, 12–14].

In this thesis, the efficiency of quantum walk search algorithms on a variety of graph

1

families will be investigated, considering both the efficiency of searching in terms of the

requisite number of steps and the efficiency of implementing the walk as a quantum circuit.

These differing notions of efficiency are linked to the symmetry of the underlying graph,

resulting in a trade-off between the benefit of quantum rather than classical search, and

the ability to efficiently implement the quantum walk. I will construct quantum circuits

implementing quantum searches along several families of graphs, in particular analysing

the search efficiency in two cases; the search for marked subgraphs of the hypercube

and the search for a single marked node of a twisted toroidal graph. In the case of the

hypercube, I prove that marked sub-m-cubes of the n-cube (where m < n) can be found

in O(√

2n) steps, generalising the results of [2].

One intriguing property of quantum walks is the level of structural information they im-

part regarding the underlying graph on which they evolve. This information is employed

implicitly in the quantum walk based search algorithms described above, and has also

been explicitly explored in a variety of other work [15–18]. Here I will explore the degree

to which the evolution of a quantum walk can characterise the underlying graph, con-

structing a quantum walk based graph isomorphism algorithm. While the work of Part II

demonstrates that this algorithm in its original form [19] does not suffice to characterise

a special family of graphs (known as k-equivalent graphs), it is able to efficiently classify

all strongly regular graphs† up to order 64, together with the additional families of graphs

considered in Chapter 4.

While the quantum walk approach to graph isomorphism testing is a recent develop-

ment, the general refinement procedure it employs relates intimately to a long-established

and well studied classical graph-theoretic technique, termed the Weisfeiler-Leman (WL)

method. In fact, I show in Part II that the quantum walk algorithm constructed in Part

I is essentially a subset of a special case of this WL method.

The WL method initially appeared to be a promising approach towards solving the graph

isomorphism (GI) problem, following a result by Babai [20,21] demonstrating that the 1-

dimensional version of the WL method suffices to characterise almost all graphs, together

with its use by [22–24] in constructing the best known upper bounds for the GI problem.

However it was shown to be insufficient to classify general graphs in Cai, Furer and

Immerman [3] and Evdokimov and Ponomarenko [4], in which families of graphs of order

n were constructed requiring O(exp(n)) time to distinguish.

†A typically difficult class of graphs to characterise, and hence often used as a test of heuristic graphisomorphism algorithms.

2

CHAPTER 1. INTRODUCTION

I analyse these ‘counterexample’ families, demonstrating that they possess common, re-

strictive properties that allow a modified version of the WL method to efficiently distin-

guish them†. In the process, various properties of the WL method are established. In

constructing the aforementioned extension to the WL method, I prove that it suffices to

characterise a particular family of graphs, to which all known counterexample graphs be-

long. In addition, there are no currently known graphs that do not belong to this family,

and I restrict the properties that such hypothetical graphs must possess if they were to

exist.

This provides renewed hope to the possibility that a simple variant of the WL method

might suffice to solve GI, and opens up new avenues in which to explore this long-standing

problem.

One of the fascinating (and useful) aspects of graph theory is its connection to almost

all areas of science. In what served as a personal reminder of this, a public lecture on

supersymmetry given by Prof. James Gates at UWA led to a collaboration investigating

the characterisation of a family of graphs called Adinkras. Developed as a graphical

encoding of representations of supersymmetry in the work of [5,25–27], Adinkras represent

a new approach to a long-standing problem in supersymmetry, the ‘off-shell problem’. This

problem essentially asks when it is possible to describe a supersymmetric theory in a way

independent to its dynamics, in other words by an off-shell representation.

The work of Gates et al. [28, 29] relates off-shell representations of supersymmetry to

a set of universal matrix algebra structures, which are in turn encoded graphically as

Adinkras. Hence in pursuing a complete classification of off-shell representations, the

classification of Adinkras is a natural step. Several works have previously dealt with this

question [25,26,30,31], succeeding in establishing many interesting properties of Adinkras.

Notably, the topological properties of Adinkras are related to doubly even codes in the

work of Doran et al. in [26] and [27].

This work classifies Adinkras via a complete characterisation of their automorphism group,

relative to the related doubly even code. In turn, the code associated with a given Adinkra

is characterised in terms of local, efficiently computable properties of the Adinkra, hence

allowing a characterisation of the isomorphism and equivalence classes of general Adinkras

relative to a small number of local parameters.

†It is expected that a similarly modified version of the quantum walk based algorithm of Chapter 4would also suffice to distinguish these counterexample graphs, as discussed in Part II.

3

Exploiting these automorphism group properties, I construct a characterisation of Adinkra

degeneracy in terms of a matrix polynomial, extending and generalising the work of Gates

et al. [32]. After formulating these results relative to a somewhat context-dependent def-

inition of isomorphism and equivalence on Adinkras (and in turn on supermultiplets), I

then consider an alternative equivalence definition, establishing the corresponding auto-

morphism group properties and in the process demonstrating that the methods of proof

used are robust to such changes.

The structure of this thesis is as follows. Part I begins by providing a formal definition of

quantum walks together with their background, properties and algorithmic applications.

This introductory material comprises Chapter 2, and contains no new work. Chapter 3,

based on our work published in [33], discusses the possibility of a practical quantum walk

algorithm providing an exponential speed-up over known classical algorithms outside of

an oracular setting. I construct quantum circuits efficiently implementing quantum walks

along several families of graphs, and present a formal proof that quantum walks can be

used to efficiently find a marked sub-cube of a hypercube. Chapter 4 completes Part I,

covering work published in [19], which details the quantum walk based graph isomorphism

algorithm discussed above. Part II focusses on what is in some sense a classical analogue of

this graph isomorphism algorithm. Introductory material regarding the Weisfeiler-Leman

method is presented in Chapter 5, after which the proposed extension to the WL method is

constructed and analysed in Chapter 6. This format is continued in Part III, which again

begins with introductory material in Chapter 7, after which the new results regarding the

characterisation of the automorphism groups of Adinkras is presented in Chapter 8. The

thesis concludes in Chapter 9.

4

Part I

Quantum Walks

5

Introduction

The considerable, ongoing interest in quantum algorithms has been sparked by the pos-

sibility of practical solutions to problems that cannot be efficiently solved by classical

computers†, following Shor’s famous prime factorisation algorithm in 1994 [6]. Further

quantum algorithms have since been constructed, notably including Grover’s search algo-

rithm, which exhibits a quadratic speed-up over the most efficient corresponding classical

algorithm [7]. However the number of known quantum algorithms is so far relatively small.

Recently the concept of a quantum walk has been introduced [1], an analogue of the clas-

sical random walk that displays many of the same properties of classical walks, along with

striking and potentially useful differences.

Classical random walks are a stochastic process, employed extensively in many fields of

science, with applications ranging from the modelling of share prices in economics [34] to

optimal search strategies [35], Monte Carlo simulations [36] and calculations of differential

equations [37]. In its simplest form, a classical random walk along a line (represented

by the set of integers) consists of beginning at a certain node, then successively flipping

a coin and moving either left or right depending on the outcome of the coin flip. The

coin provides a means of randomising the chosen directly of propagation. In a simple

unbiased walk such as this, at each step there is equal probability of moving either left

or right, and hence the position probability distribution at some number of steps can be

modelled by a binomial distribution. After t steps, for sufficiently large t, the probability

distribution can be approximated by a Gaussian with standard deviation√t. Note that

the classical walk model described above is a two-step process, namely flipping a coin - the

‘coin operation’, then shifting the walker based on the outcome - the ‘shifting operation’.

It is not reversible; in essence, the state of the coin is reset after each step.

First proposed in 1993 by Aharonov et al. [1], quantum walks are instead a unitary process,

†Or at least problems for which there are no known efficient classical solutions.

6

and thus both reversible and deterministic, in which the outcome of the coin operation

depends on the outcome of previous coin flips. Rather than a single possible path be-

ing sampled in a given iteration of a classical random walk, the evolution of a quantum

walk is modelled by a probability amplitude distribution. These distinctions result in

markedly different properties, from which non-intuitive dynamics often emerge. For ex-

ample, returning to the example of a simple unbiased walk on a line, the resulting position

probability distributions of classical and quantum walks after a fixed number of steps are

contrasted in Fig. 1.1. As opposed to the Gaussian distribution of the classical random

walk, the quantum walk on a line exhibits interference fringes, and displays quadratically

faster mixing times. Similarly, the quantum and classical walks on a 2-dimensional grid

are contrasted in Fig. 1.2. Again the Gaussian distribution of the classical random walk

is markedly different to the spreading characteristics of the quantum walk.

-100 -50 0 50 1000.00

0.02

0.04

0.06

0.08

Position

Prob

abili

ty

Figure 1.1: Probability distributions resulting from unbiased classical (dashed) and quan-tum (solid) walks on Z.

Figure 1.2: Probability distributions resulting from unbiased classical (left) and quantum(right) walks on a 2-dimensional grid. Here the horizontal axes represent position, and thevertical axis represents probability.

Indeed, a general, useful characteristic of quantum walks is a propensity for quadratically

7

faster hitting and mixing times along highly symmetric graphs, when compared to classical

random walks. Given the extensive algorithmic applications of classical random walks, and

the usefully distinct properties of quantum walks, together with their in some sense non-

intuitive dynamics, it is hoped that quantum walks may similarly provide a fruitful source

of novel quantum algorithms. Several algorithmic applications of quantum walks have

already been developed [2,8–11,16,17,19,38–40], and a brief overview of these is given in

Section 2.2.

In this Part, a broad but brief overview of quantum walks is presented, together with some

recent results in Chapter 2. In particular, we give a background to quantum walk search

algorithms and quantum circuits. This leads into the results of the following chapter,

in which we discuss the possibility of a practical quantum walk algorithm providing an

exponential speed-up over known classical algorithms outside of an oracular setting. In

the context of introduced definitions of efficiency relating to both circuit implementations

and quantum search, we present several families of graphs which allow efficient quantum

search relative to both of these metrics, constructing explicit quantum circuits for each

case. New results involving search for marked subgraphs of hypercubes and quantum

search along twisted toroids are presented. In particular, it is shown that a marked sub-

cube of a hypercube with N vertices can be identified using O(√N) calls to an oracle,

generalising the results of Shenvi et al. [2].

Finally, Chapter 4 presents novel quantum and classical algorithms relating to the graph

isomorphism problem. This approach to the graph isomorphism algorithm is tested against

an extensive database of graphs, including all strongly regular graphs up to order 64. All

tested graphs are successfully distinguished. The quantum algorithm employs a novel

measurement method to directly compare the amplitude distributions related to each

graph (of order n) using a single measurement. Any differences existing between these

amplitude distributions can be distinguished up to an arbitrary high probability with this

single measurement, via extending the related quantum walk by O(log(n)) steps.

8

Chapter 2

Properties of Quantum Walks

Similarly to the classical case, there are two broad classes of quantum walks: discrete-

time and continuous-time. In this work we focus only on discrete-time quantum walks -

for extensive reviews of both continuous-time and discrete-time quantum quantum walks

see [41–43]. Whilst in many cases their properties coincide [44, 45], some of the algorith-

mic applications are phrased relative to continuous-time quantum walks [9, 46], and they

provide an alternative framework for understanding the underlying dynamics.

2.1 Formal Definition

Here we provide a general definition of a quantum walk along an undirected graph G(V,E),

with vertex set V = v1, v2, . . . and edge set E = vi, vj, vk, vl, . . . consisting of

unordered pairs of connected vertices. The neighbours of a given vertex v ∈ V are denoted

by

d(v) = x ∈ V : v, x ∈ E, (2.1)

where dv := |d(v)| is termed the valency of v. The quantum walk acts on an extended po-

sition space, in which each node v with valency dv is split into dv subnodes, corresponding

to the neighbours of v. Specifically, the subnodes of a vertex v ∈ V are represented by the

set d(v). This position Hilbert space is represented by

H = HP ⊗HC , (2.2)

9

2.1. FORMAL DEFINITION

where HP and HC represent the set of vertices and di-edges (alternatively the endpoints of

edges) of G respectively. Explicitly, HP is termed the position space, with dimension |V |,spanned by the set of orthonormal basis states |v〉 : v ∈ V . HC is termed the coin space,

with dimension bounded by the maximum valence of G, and for each vertex v ∈ V it is

spanned by |cx〉 : x ∈ d(v). Hence H is spanned by the set of states |v, cx〉 := |v〉 ⊗ |cx〉,where v ∈ V and x ∈ d(v).

A step of a quantum walk on G is described by the unitary operator

U = S · C. (2.3)

The action of the shifting operator S is defined by

S|v, cx〉 = |x, cv〉, (2.4)

for all v, x ∈ E. The coin operator C comprises a set of unitary operations Cv, termed

coins, one for each vertex v ∈ V . Each Cv can be represented by a unitary (dv×dv) matrix

which mixes the dv probability amplitudes associated with v. Throughout this work we

consider predominantly unbiased coin operators - those for which the constituent coins are

invariant to permutations of the subnode labels. In particular, in situations where a fixed

or canonical labelling of the vertices is not known, an unbiased coin operator (composed

of unbiased coins) must be used to maintain consistency and reproducibility of results.

Example 2.1.1. Consider a simple† graph G(V,E) with |V | = n vertices and |E| = k

edges. Then the basis for the Hilbert space H on G is a state space containing 2k states.

The shifting operator S can be represented by a (2k × 2k) permutation matrix, and

grouping the coin states by common vertex yields a coin operator C represented by a

(2k × 2k) block diagonal unitary matrix.

Since k has an upper bound of n(n−1)/2, it follows that a step of the walk, U = S ·C, can

be simulated efficiently on a classical computer, in a time that scales with O(n6). In fact

since S and C are represented by permutation and block diagonal matrices respectively,

this can be trivially reduced to O(n4).

Example 2.1.2. Commonly used unbiased coins include: the Grover coin (related to the

†A simple graph is a graph containing no self-loops or multiple edges.

10

CHAPTER 2. PROPERTIES OF QUANTUM WALKS

Grover search algorithm [7,38]), defined on d states as

(Gd)i,j =2

d− δi,j , (2.5)

where δi,j is the Kronecker delta, and the identity operator 1. The addition of a phase

factor eiπθ to a coin that is unbiased preserves this property, hence for example the negative

identity is also unbiased in this sense.

Note that it is the interference between the complex amplitudes of the quantum walk as

it evolves that leads to its quantum nature, producing for example the interference peaks

of Fig. 1.1.

2.2 Algorithmic Applications of Quantum Walks

In this section we briefly describe some of the major results stemming from quantum

walk research, with a focus on algorithmic applications, and in particular, quantum search

via quantum walks. For both experimental and theoretical results pertaining to physical

implementations of quantum walks, see for example [47–51]. A brief discussion of quantum

circuits is also included, as the focus here will be on quantum walks implemented via a

universal set of elementary quantum gates.

2.2.1 Quantum Oracles

The significance of most quantum walk algorithms discussed in this section lies in a prov-

able speed-up over possible classical algorithms. It is possible to prove such speed-ups,

however the only such cases constructed up to this point involve a speed-up relative to a

given oracle - in an oracular setting.

An oracle can be thought of as a black box subroutine available to an algorithm, performing

some particular function in a single step. Typically an oracle will perform some kind of

Boolean function,

f : Zk2 → Z2, (2.6)

for instance deciding membership of an input bit-string within some list. The complexity

of an algorithm solving some problem relative to a particular oracle is determined by the

11

2.2. ALGORITHMIC APPLICATIONS OF QUANTUM WALKS

number of queries or calls the algorithm must make to the oracle before halting. Oracles

are often used in computational complexity science to define various complexity classes.

In the case of a quantum algorithm, we assume that calls to an oracle can be made in

superposition - such a situation is also referred to specifically as a quantum oracle.

The usefulness of oracles in developing quantum walk algorithms is that within an oracular

setting the following types of strong results can be proven:

• Algorithms can be shown to be optimal - in other words, no algorithms can possibly

solve the problem with fewer oracle queries.

• The optimal complexity of certain subclasses of algorithms can be determined. For

instance, the complexity of optimal quantum algorithms and optimal classical algo-

rithms can be separated.

This concept underlies all computational speed-ups demonstrated using quantum walks -

they allow a problem to be solved faster than any possible classical algorithm, relative to

a particular oracle.

Note that while incredibly useful from a computational complexity viewpoint, in practical

algorithms the action of oracles also needs to be implemented. Hence while oracular

settings provide important insights into different models of computation, in cases where

the oracle itself cannot be implemented efficiently, a separation between computation

models relative to this oracle may be of limited value.

2.2.2 Faster Hitting Times

As mentioned previously, one of the properties separating quantum and classical walks is

hitting and mixing times on various classes of highly symmetric graphs. Several definitions

of the term hitting time can be constructed; here we consider the definitions of hitting

time provided in Kempe [52] and Venegas-Andraca [53].

Definition 2.2.1 (One-shot hitting time). If a quantum walk beginning in state |x〉 is in

state |y〉 after T steps with probability greater than or equal to p, then the walk is said

to have a (T, p) one-shot (y, x) hitting time.

Definition 2.2.2 (|y〉-stopped walk from x). Consider a walk beginning in state |x〉 in

which after each step the measurement of the state |y〉 is made. If the walk is found in

this state, the |y〉-stopped walk is halted at this step, otherwise it continues.

12


Definition 2.2.3 (Concurrent hitting time). A quantum walk has a (T, p) concurrent

(y, x) hitting time if the |y〉-stopped walk from x stops at or before T steps with probability

≥ p.

Although these two definitions of hitting times are usefully distinct, and presuppose very

different measurement methods, in most cases regardless of which metric is used the algo-

rithmic speed-up is the same.

Probably the best known example of an algorithmic speed-up in hitting time via quantum

walks is provided by the work of Childs et al. [10] in their paper regarding quantum walks

on glued trees. A family of glued trees is constructed which quantum walks can traverse

exponentially faster than any possible classical method, relative to a particular quantum

oracle. This algorithm, and in particular the oracle used, is investigated in more detail in

Chapter 3.

2.2.3 Search Algorithms

Related to faster hitting times is the search problem, currently one of the most active

research areas relating to algorithmic applications of quantum walks. The general problem

can be described as follows. A graph G(V,E) is given, in which some set of nodes M ⊂ Vare marked, where |M | ≥ 1. The goal is to find some or all of the marked nodes.

This is generally accomplished via queries to an oracle which essentially stores the position

of the marked node(s), for instance an oracle performing the function

f(x) =

1 if x ∈M,

0 otherwise.

If |M | is sufficiently small, a classical algorithm which does not exploit some special struc-

ture of G † requires O(|V |) calls to the oracle to find a marked node with sufficiently high

probability [54]. Conversely, Grover’s algorithm [7], also implemented by quantum walks

by Shenvi et al. [2], allows quantum search of an unstructured database (or a highly sym-

metric graph in the case of quantum walk algorithms) of size n with only O(√n) queries

to the oracle.

The idea of quantum search on graphs (as opposed to the unstructured search of Grover’s

algorithm) was first considered in the context of search on a d-dimensional hypercube (or

†For instance, if the marked node(s) are in some sense randomly placed in the graph.

13


d-cube) in the work of Shenvi et al. [2], and Aaronson and Ambainis [54]. In these works,

search for a single marked node on a d-cube with n = 2d vertices was shown to require

O(√n) queries.

Several quantum walk search algorithms have since been constructed, involving several

classes of graphs, in all cases either highly symmetric or having bounds on their complexity

imposed by conditions such as sparsity. Classes of graphs involved include d-dimensional

grids [9, 12], complete graphs [13, 14], hypercubes [2] and sparse graphs [18]. Interesting

recent results from Childs and Kothari [18] demonstrate novel techniques for finding certain

marked subgraphs of graphs with a bounded number of edges. The techniques they use

differ significantly from prior quantum walk search algorithms, for instance with queries

to the oracle being used to decide which vertices of the graph to search over.

Almost all such algorithms use the technique of amplitude amplification, in which the

quantum walk is first applied until the probability of finding the walker at the marked

node is ‘sufficiently high’ - some value given by ε. Upon iterating these set of steps of the

quantum walk O(1/√ε) times, with an appropriate phase shift after each iteration, the

walk is found at the marked node with probability Ω(1) [55,56].

Example 2.2.4. A general framework for quantum search along a graph is given here,

developed in [2,9,57]. Here the graph will be considered to be a complete graph, although

the same method can be applied to several other families of graphs, and is detailed in

Chapter 3. Given a complete graph G on n nodes containing a single marked node xtarget,

the walk is initialised in an equal superposition of all states. A quantum oracle O is

available, performing the function

O(x) =

1 if x = xtarget,

0 otherwise.

Rather than the standard Grover coin described in Eq. (2.5), a perturbed Grover coin is

applied at each step of the walk, in which the action at the marked node is instead the

negative identity operator. The marked node can be found after O(√n) queries of the

oracle [57].

2.2.4 Element Distinctness

Given a list of N bit-strings, the element distinctness problem is to determine whether

or not all the strings are different. In other words, it is to find two bit-strings that are

14


the same, if any such pairs exist. Extending a general quantum algorithm for spatial

search constructed by Aaronson and Ambainis [54], the work of Ambainis [8] constructs

a quantum walk algorithm solving the element distinctness problem, by first converting

the list of bit-strings into a graph with marked vertices. Relative to a given oracle, this

algorithm solves the element distinctness problem using O(N2/3) queries, compared to the

fastest possible classical algorithm requiring Ω(N) queries. This O(N2/3) algorithm has

also been shown to be optimal [58,59].

2.2.5 Graph Isomorphism

Discussed in detail in both Chapter 4 and throughout Part II, the graph isomorphism

problem consists of determining whether or not there exists a bijection between the vertex

sets of two given graphs that preserves their edge sets; in other words whether or not

they are isomorphic. Several quantum walk methods have been proposed for tackling this

problem, or subsets of the problem [16,17,19,60–62], and these will be covered in Chapter

4.

2.2.6 Quantum Simulation and Universal Quantum Computers

The simulation of quantum systems has long been one of the primary goals of quantum

computing algorithms, since it was proposed by Feynman in 1982 [63]. Following the work

of Lloyd [64], demonstrating that local quantum systems can be efficiently simulated using

a quantum computer, a few works have focussed specifically on quantum simulation using

quantum walk algorithms. Examples of this include the work of Berry et al. [65], in which

they demonstrate a method for efficiently simulating the evolution of sparse Hamiltonians

using quantum walks. Similarly, in [45] an alternative method is developed by Childs,

demonstrating connections between the continuous and discrete-time quantum walks, and

providing a method of efficiently simulating Hamiltonians in certain non-sparse cases using

quantum walks.

The interference properties of quantum walks are linked to the one-dimensional Dirac

equation by Strauch in [66], showing that the time evolution of the quantum walker,

initially localised on a lattice, is directly analogous to relativistic wave-packet spreading.

This work is extended by Bracken et al. in [67], in which it is shown that the probability

density of a free Dirac particle at any given time can be obtained asymptotically from the

15


probability distribution of a quantum walk. An explicit physical scheme for implementing

such a simulation using ultra-cold spinor atoms in driven optical lattices is proposed by

Witthaut in [68].

Finally, one further possible application of quantum walks which has recently arisen is that

of universal quantum computation. While perhaps not the most intuitively natural system

of computation, in Childs [39] continuous-time quantum walks are shown to yield a scheme

for universal quantum computation. Specifically, it is shown that a Hamiltonian restricted

to being the adjacency matrix of a low-degree graph is sufficient to span BQP - in other

words that any quantum algorithm can be recast in terms of such Hamiltonians. Since

such a Hamiltonian can be efficiently simulated by continuous-time quantum walks, they

are able to implement a universal gate set. This work was also extended to discrete-time

quantum walks by Lovett et al. [40] and Underwood and Feder [69].

2.2.7 Quantum Circuits

Chapter 3 focusses specifically on quantum walks implemented via a universal set of el-

ementary quantum gates, with algorithms described by specific quantum circuits. Some

requisite background will be briefly introduced in this section.

A universal quantum computer is a specific model of computation that can express any

quantum algorithm. A quantum circuit is one example of such a model; as mentioned

above, quantum walks are another such model [39]. Similarly a set of elementary quantum

gates (or universal quantum gates) are a set of unitary operators that can be combined to

form a quantum circuit that implements any quantum algorithm. The term elementary

generally implies that these gates act on only a small number of qubits, forming the

building blocks for larger operations, analogous to classical logic gates.

One such commonly used universal set of quantum gates comprises three gates; the 2-qubit

controlled-NOT gate, or CNOT gate, and two 1-qubit gates generating U(2), for instance

the phase and Hadamard gates. A matrix representation of the CNOT gate is given by

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

,

with the standard notational convention such that the rows and columns denote from

16


left to right and top to bottom respectively the two-qubit states |00〉, |01〉, |10〉 and |11〉.Representations of the phase and Hadamard gates are given by

φ =

1 0

0 eiπφ

and

H =1√2

1 1

1 −1

respectively.

Rather than phrase quantum circuits in terms of only these few gates (although it is

possible since they are a universal gate set), a larger set of gates are often used for the

purposes of brevity. We introduce these gates and their circuit diagram symbols below,

via a set of circuit diagram examples. Note that the convention to be used here in circuit

diagrams is that the wires of the circuits represent the qubits, with the top-most qubit

corresponding to the left-most bit in the related bit-string. Circuits proceed from left to

right.

The circuit of Fig. 2.1 below presents a set of 5 gates, illustrating the gate symbols to be

used in this work.

G |1〉 •

× • ×× •

Figure 2.1: An example quantum circuit, demonstrating the standard gates and conven-tions used.

The |1〉 preceding the second qubit indicates that this qubit is initialised in this state. The

first gate, acting on only the fourth qubit, is the NOT gate, implementing the operation

0 1

1 0

.

17


The second gate is the SWAP gate, implementing the operation

1 0 0 0

0 0 1 0

0 1 0 0

0 0 0 1

.

The third gate is the CNOT gate described above, implementing a NOT gate on the target

qubit depending on whether the control qubit is in state |0〉 or |1〉, indicated by a white

or black circle respectively. In this case the white circle indicates that the NOT gates is

implemented if the control qubit is in state |0〉, hence this gate implements the matrix

0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1

.

The final two gates are generalised controlled gate operations. In the first case, the gate

operation referred to here is the Grover operator acting on the first qubit, with the action

of this gate dependent on the state of the other three qubits as indicated. Similarly in the

last gate the controlled operation performed is the NOT gate, acting on the fourth qubit

dependent on the state of the first three. This is also known as the generalised CNOT

gate, or CnNOT gate given n control bits. In Tucci [70] an efficient implementation of

generalised controlled gate operations are given, requiring O(n2) elementary 2-qubit gates

to implement.

18

Chapter 3

Quantum Walk Based Search

Algorithms

Based on B. L. Douglas and J. B. Wang’s original publication in Phys. Rev.

A 79, 052335 (2009) [33] (with revisions in Phys. Rev. A 80, 059901E

(2009) [71] and also featured in the June 2009 issue of Virtual Journal of

Quantum Information) together with subsequent currently unpublished work.

The original context has been significantly expanded, including an extension

of the quantum circuits to generalised cases, additional analysis of search

complexity characteristics, the inclusion of additional graph families and

further supporting material.

3.1 Introduction

The considerable, ongoing interest in quantum algorithms has been sparked by the pos-

sibility of practical solutions to problems that cannot be efficiently solved by classical

computers. In other words, the opportunity to achieve exponential speed-ups over clas-

sical techniques by harnessing entanglement between densely encoded states in a quan-

tum computer. Quantum walks have been the focus of several recent studies (see for

example, [41, 72–75]), with particular interest in possible algorithmic applications of the

walks [2, 8, 10, 43, 76]. A few such algorithms have already been developed, perhaps the

most notable being the ‘glued trees’ algorithm developed by Childs et al. [10], in which

quantum walks are shown to traverse a family of graphs exponentially faster than any

19

3.2. METRICS OF EFFICIENCY

possible classical algorithm, relative to a particular quantum oracle.

Along certain types of graphs, in particular graphs whose global structure can be charac-

terised by a small number of parameters, quantum search algorithms have been shown

to yield a quadratic speed-up over their classical equivalents, relative to a fixed ora-

cle [12, 77, 78]. These types of graphs include sparse graphs with efficiently computable

neighbours [79], and various families of highly symmetric graphs [13,33].

Previous studies of quantum walk based search algorithms have focussed on a computa-

tional complexity comparison between the quantum search and the best possible classical

search along a particular graph. This requires comparing the relative number of queries

to a fixed (quantum) oracle needed to complete the search. As such, there has been no

reason to consider the resource requirements of the oracle itself.

However, an efficient practical implementation of such a search algorithm would require

an efficiently implementable oracle. This in turn requires the ability to efficiently perform

steps of the quantum walk along the graph. So far efficient quantum search has been

shown to be possible only along certain types of graphs [12, 13, 38, 77] belonging to the

class mentioned above, whose global structure can be completely characterised by a small

number of parameters. Several graphs belonging to this category will be considered here,

and shown to be amenable to exact, efficient quantum circuits implementing quantum

walks along them.

In this chapter we discuss the possibility of a practical quantum walk algorithm providing

an exponential speed-up over known classical algorithms outside of an oracular setting -

in other words without the use of an oracle. We define notions of efficiency with respect to

both circuit implementations of quantum walks and quantum walk search algorithms, and

present several families of graphs allowing efficient quantum search relative to both of these

metrics. For these families, quantum circuits are constructed that explicitly implement

the related search algorithms. In particular, we provide a numerical analysis of a quantum

walk based search along a twisted toroid family of graphs, which indicates that O(√n)

steps of the walk are sufficient to find the marked node.

3.2 Metrics of Efficiency

Discussions of efficiency in the preceding section refer to two distinct situations: the

efficiency of a search algorithm, measured by the required number of oracle queries, and

20

CHAPTER 3. QUANTUM WALK BASED SEARCH ALGORITHMS

the efficiency of the implementation of a single step of the quantum walk. In the situation

pertaining to all currently developed search algorithms, these are independent metrics,

in that the ability to implement a single step of the walk efficiently does not effect the

required number of oracle queries.

Explicitly, given a quantum walk along a graph on n vertices, we define an efficient quantum

walk to be one in which a single step is implemented using O(polylog(n)) elementary (2-

qubit) gates. Conversely, we define an efficient quantum search algorithm to be one which

exhibits at least a quadratic speed-up over the best possible classical search, relative to

a particular oracle. Outside of an oracular setting we simply define an efficient search to

be one which requires O(√n polylog(n)) := O(

√n) elementary quantum gates to find the

marked node. Note that this definition stems from the O(√n) lower bound on unstructured

search [80] combined with the metric for an efficient quantum walk step of O(polylog(n)).

Which of these two metrics applies when the term efficiency is used should be clear from

the context.

3.3 Efficient Circuit Implementation

Throughout this chapter we follow the quantum walk formalism developed in Chapter

2. Regarding the possibility of a quantum walk algorithm yielding an exponential speed-

up over the best known classical algorithm, note that quantum walks themselves can be

simulated classically in polynomial time, scaling with graph size. So to allow even the

possibility of such exponential speed-ups, quantum implementations of the quantum walk

must scale logarithmically with graph size, satisfying the notion of efficiency above.

Many of the currently proposed “natural” physical implementations of quantum walks [49,

81–83] cannot achieve this, as the walks evolve on nodes that are implemented by physical

states, on which operations are directly performed. Hence the resource requirements grow

at best polynomially with the state space. In order for an exponential gain to be possible,

the nodes must be encoded by a string of entangled states, such as qubits in a quantum

computer, making use of memory that grows exponentially with the number of qubits.

In addition, the number of elementary gates required to perform the walk must grow

logarithmically with the size of the state space.

So far, this has only been found to be possible for structures with a high degree of sym-

metry - where symmetry in this case refers to the ability to characterise the structure by a

21

3.3. EFFICIENT CIRCUIT IMPLEMENTATION

small number of parameters, increasing at most logarithmically with the number of nodes.

Note that this may not necessarily imply that the structure has geometric or combinatorial

symmetry in the typical sense of the terms. For instance, sparse graphs with efficiently

computable neighbours fall into this category, and as a consequence of [65, 79] have been

shown to allow efficient implementations of quantum walks. Here sparse graphs of order n

are defined as in [65] to have degree bounded by O(polylog(n)), with the further condition

that the neighbours of each vertex are efficiently computable. Possessing efficiently com-

putable neighbours implies the existence of an O(log(n)) sized function characterising the

graph, such that the information contained in the O(n) edges can be compressed to size

O(log(n)). This compression seems to require the presence of some kind of structure to

the system, for example, the graph cannot contain more than O(log(n)) randomly placed

edges. An interesting open question is whether the automorphism group of sparse graphs

with efficiently computably neighbours must be non-trivial.

3.3.1 Circuit Diagrams

When constructing explicit quantum circuits implementing quantum walk search algo-

rithms, we make use of several non-standard quantum gates for the purposes of simplifying

the circuit diagrams. Below we define these quantum gates, and demonstrate that they

can be efficiently implemented using elementary 2-qubit gates. Background information

regarding general quantum circuit conventions and notation can be found in Nielsen and

Chuang [84] and Aharonov [85].

Firstly, consider a general permutation of a set of t qubits. Any such permutation can be

implemented by a series of at most O(t) transpositions (i.e. 2-qubit swap operations). In

particular, we consider a rotation of the qubit labels, such that the ith qubit is mapped

to the (i± 1)th qubit, denoted by the operations RR and RL, or right and left rotations.

These rotations are implemented via the quantum gates:

× × × ×× ×

......RL =

×RR =

×× × × ×× × × ×

22


The qubit ordering is such that given a t-qubit state ~x = (xtxt−1 . . . x1), we have

RL(~x) = (xt−1xt−2 . . . x1xt), and

RR(~x) = (x1xt . . . x2).

Note that the convention used here dictates that the tth qubit in this case is represented

by both the left-most bit in the bit-string and the upper-most qubit in the circuit diagram.

Similarly to a permutation of the qubit labels, we also construct quantum gates represent-

ing permutations of the 2t possible t-qubit states. Clearly not all such permutations can

be efficiently implemented; here we consider only rotations and transpositions.

Left and right rotations of t-qubit states, denoted increment and decrement operators

respectively, have the following action. Given an t-length bit-string ~x, we define

incr(~x) = ~x+ 1, decr(~x) = ~x− 1,

performing a cyclic permutation (in either direction) of the n-qubit states, and imple-

mented by the following quantum gates:

...

...incr. = • decr. =

• • • • • × ×

Note that although the above gates perform a cyclic rotation by a single state, the same

procedure can be extended to perform a rotation by any number of states, by first decom-

posing it into a series of rotations of size 2m for a set of integers m < t. For instance, an

incremental rotation of seven states applied to the 32 states represented by five qubits can

be achieved by applying 3 successive rotations of 4, 2, and 1 states, as explicitly shown in

Fig. 3.1, rather than applying the increment operator 7 times.

• • • • • × • • • •

• • • × • • • • • • • ×

Figure 3.1: A rotation of seven states, split into the composite powers of 2, being threerotations of size 4, 2 and 1 states respectively.

23

3.3. EFFICIENT CIRCUIT IMPLEMENTATION

Note that the operators permuting both the qubit labels and t-qubit states defined above

can be implemented using at most O(t3) elementary 2-qubit gates. Specifically, the per-

mutations of qubit labels require only O(t) 2-qubit swap gates, whereas the increment and

decrement operators require O(t) generalised CNOT gates (or CtNOT gates), which in

turn can be implemented by O(t2) elementary 2-qubit gates.

Transpositions between t-qubit states can similarly be implemented usingO(t2) elementary

2-qubit gates. Specifically, any two states differing by exactly m qubits can be transposed

using 2m + 1 generalised CNOT operations. For example, given 16 states encoded by 4

qubits, the lexicographically 1st and 10th states (represented by |0000〉 and |1001〉 respec-

tively) can be transposed via 3 controlled swap operations, as shown in Fig. 3.2. Using

this method any transposition of states on t qubits can be performed using a maximum

of 2t− 1 generalised CNOT gates, or 2t2− 3t C2NOT gates. This may not be the optimal

way to implement a particular transposition, however it does scale logarithmically with

the number of states.

• • Figure 3.2: A transposition of the |0000〉 and |1001〉 states.

Using similar methods, other permutations with essentially binary characters can also be

efficiently implemented, such as swapping every second state, or performing some given

internal permutation to each consecutive group of 8 states (or 2m states, for some fixed

integer m). Note that permutations which may not seem to have a binary character can

be transformed to efficiently implementable permutations. For instance, if we wished to

split the set of states into groups of 6, and swap every 4th element, we could achieve this

by expanding the state space - embedding each group of 6 into a group of 8, with the last

two states remaining unused, ‘empty states’.

For simplicity, given an implementation on qubits, the preceding examples have all been

essentially binary in nature. Efficient implementations using qubits are equally possible on

other, non-binary structures, such as ternary trees or complete 3t graphs, although they

will be restricted to approximate rather than exact implementations. For example, imple-

menting the complete 3t graph (with self-loops) using a qubit circuit requires many more

2-qubit gates, given the need to approximate a 9D Hadamard or Grover coin operator over

16 states, without mixing into the other 7 states. As would be expected, a more natural

24


implementation is possible if qutrits are used instead. In this case, the coin operator is

again nearly separable if using the Grover coin operator, and completely separable if using

a qutrit equivalent of the Hadamard operator. Here we take a qutrit equivalent of the

Hadamard operator to be an operator Tt acting on t qutrits, satisfying:

((T1)±)a,b =1√3e±i

2π3a b, where a, b ∈ 0, 1, 2, and

(Tt)± = (T1)± ⊗ (Tt−1)±.

Qutrit circuits implementing a quantum walk along the complete-3t graph using the T coin

operator or the Grover coin operator can then be constructed as in Fig. 3.3. Nevertheless,

the use of a more natural base provides at best a polynomial efficiency gain.

×

...

×node

×

T+ T− ×

T+ T−

...

×subnode

T+ π T− ×

|0〉 •

Figure 3.3: Qutrit-based quantum circuit implementing a quantum walk along a complete3t-graph.

3.4 Efficient Search

In this section we briefly note known types of graphs which are amenable to efficient search

via quantum walks, in the sense described in Section 3.2. Although nominally independent

to the notion of efficient circuit implementation, the known families of graphs in each case

overlap significantly.

As for efficient circuit implementation implementation, the efficient search algorithms

which have been developed to this point act exclusively on graph which are either highly

symmetric or whose global structure can be completely characterised by a small number of

25

3.5. SPECIFIC EXAMPLES

parameters. For example, these families include Johnson graphs [8], Hadamard graphs [18],

sparse graphs [18, 65, 79] as well as hypercubes and hypercycles, also discussed in the

following chapter. The general method of the majority of quantum walk search algorithms

follows that described in Chapter 2, involving a perturbation in the coin operator at the

site of the marked node(s), alternatively implemented by either the oracle or the quantum

walker.

3.5 Specific Examples

In this section, we give examples of a few families of graphs for which relatively simple

quantum circuits can be designed to efficiently implement quantum walks along them.

Additionally, these will be graphs yielding efficient quantum walk search algorithms. For

most such families, explicit quantum circuits will be constructed implementing these search

algorithms. In the remaining cases the quantum circuits constructed will simply implement

a single step of a quantum walk, without regard to a search algorithm. The goal is to

identify families of graphs which can be efficiently walked along, and to note common

properties held by these types of graphs. The discussion of search algorithms will be

predominantly cursory in nature, stemming from previous literature results in most cases.

The exception to this is the twisted toroid and glued tree families of graphs of sections

3.5.4 and 3.5.5 respectively. In the case of the twisted toroids, we provide a numerical

analysis of a corresponding quantum walk search algorithm, indicating that on n vertices,

O(√n) steps of the walk are sufficient to find the marked node.

3.5.1 Cycle

Firstly, we look at a simple cycle, considering only the implementation of a quantum

walk along it. To implement such a quantum walk, we first note that each node has two

adjacent edges, and hence two subnodes†. Proceeding systematically around the cycle, we

assign each node a bit-string value in lexicographic order, such that adjacent nodes are

given adjacent bit-strings. Hence for a cycle of order 2t, t qubits are required to encode

the nodes, and an additional qubit to encode the subnodes. The coin operation can be

implemented by a single Hadamard gate acting on the subnode qubit, and the shifting

operation by a cyclic permutation of the node states, in which each state (or bit-string) is

†Recall that the ‘subnodes’ refer to the coin states of a given node, each essentially corresponding tothe endpoint of an edge incident with that node.

26


mapped to an adjacent state (either higher or lower depending on the value of the subnode

qubit).

This permutation can be achieved via ‘increment’ and ‘decrement’ gates, as described in

Section 3.3.1, made up of generalised CNOT gates. These gates produce cyclic permuta-

tions (in either direction) of the node states. The resulting shifting operator is

S = (incr.⊗ |1〉+ decr.⊗ |0〉).

Here the tensor space description separates the node and subnode states. So to implement

a walk along a cycle of size 2t we require t + 1 qubits. The number of elementary gates

required is hence limited to O(t), and so both memory and resource requirements scale

logarithmically with graph size. An example of the circuit for a cycle of size 16 is given

in Fig. 3.4, with the qubits encoding the ‘node’ and ‘subnode’ states labelled. Note that

although this specific implementation requires a cycle of order 2t, only trivial alterations†

are required to efficiently implement cycles of any size. For instance, an equivalent circuit

for a cycle of size 25 is given in Fig. 3.5.

incr decrnode

subnode H •

Figure 3.4: Quantum circuit implementing a quantum walk along a 16-length cycle.

incr

decr

• • •• • • • •

node • • • • • • •

subnode H • • • • • •

Figure 3.5: Quantum circuit implementing a quantum walk along a 25-length cycle.

†In fact such alterations correspond precisely to the addition of transposition operators as discussed inSection 3.3.1.

27


3.5.2 Hypercube

A similar method can be used to efficiently implement a walk along a t dimensional grid

or hypercube, by partitioning the labels of the nodes into t distinct sets corresponding

to each coordinate. Consider specifically the t-cube (or t-dimensional hypercube), with

n = 2t nodes, each of valency t. The graph for t = 4 is shown in Fig. 3.6. Shenvi

et al. [2] demonstrated an efficient quantum search algorithm for finding marked nodes

on the hypercube. Their algorithm operates quadratically faster than classical methods,

requiring O(√n) calls to an oracle that effectively acts as a coin operator which is biased

only relative to the marked node. Specifically, the oracle can be viewed in two ways. It

can perform the operation of Eq. (2.6), assigning (via some ancillary qubit) a 1 to the

marked state and a 0 to the unmarked states. Or alternatively it can perform the entire

coin operation, acting with the Grover coin on the unmarked nodes and with the negative

identity operator on the marked node.

Figure 3.6: The 5 dimensional hypercube.

Here we show that this oracle can be implemented efficiently, due to the symmetric nature

of the graph, requiring O(log(t)) elementary 2-qubit gates per call. The coin operator

C will be defined as C = G ⊗ (1 − |x〉〈x|) + C′ |x〉〈x|, where C ′ is the perturbed coin

acting on the marked node x (for the examples presented here, C ′ = −1). G is the Grover

operator, defined on d dimensions by Gi,j = 2d−δi,j . The shifting operator has its standard

definition, provided in Eq. (2.4) above, such that for u, v ∈ E, S|u, cv〉 = |v, cu〉, where

S2 = 1.

An example quantum circuit implementing a step (U = S · C) of the walk along the t-

dimensional hypercube is shown in Fig. 3.7. Here the G operator represents the Grover

operator defined above, and the π operator represents a π phase change applied to each

qubit it acts upon (i.e. the negative identity operator). This circuit requires t ‘node’ qubits,

28


with each bit string representing a node of the hypercube, with the natural ordering on

the hypercube, such that two nodes are connected if and only if their bit strings have a

Hamming distance of 1. Log(t) ‘subnode’ qubits are required (rounded up to the nearest

integer), with the first 2t bit strings representing the 2t edges of each node. Finally, O(t)

auxiliary qubits are required (although not shown) to implement the CtNOT gates, to be

discussed in more detail below. Note that the first three operations shown on Fig. 3.7

implement the perturbed coin operator C ′, while the subsequent generalised CNOT gates

implement the shifting operator.

• •

node ......

G G π

• •... · · · ...subnode • • • •

Figure 3.7: Quantum circuit implementing a step of the quantum search along the t-dimensional hypercube, with n = 2t nodes. The circuit contains t ‘node’ qubits and log(t)‘subnode’ qubits, and the single marked node is represented by the state |01 · · · 00〉.

This circuit provides an efficient implementation, in that O(log(n)) elementary 2-qubit

gates are required per step. The Grover operator G acting on the log(t) subnode qubits

requires O(log(t)) 2-qubit gates [86]. The generalised conditional NOT operations are

described by Tucci [70], in which it is shown that a generalised CtNOT gate, performing

a NOT operation conditionally based on the state of t other nodes, can be implemented

using O(t) 2-qubit gates together with O(t) auxiliary qubits. Performing both these and

the conditional G and π operations requires O(t) 2-qubit gates and an additional O(t)

auxiliary qubits (see, for example, [70, 86]).

Hence the circuit of Fig. 3.7, performing a step of the walk implementing a quantum

search along the hypercube on t-dimensions (with n = 2t nodes), can be implemented

using O(log(n)) qubits and O(log(n)) 2-qubit gates. This circuit is a representation of

a single step of the search algorithm of Shenvi et al. [2], with the explicit action of the

oracle included in the circuit. For the 5-dimensional hypercube, given an initial equal

superposition across all states, the resulting probability distribution against number of

29


steps is given in Fig. 3.8. Hence as in [2], after O(√n) steps of the walk (or equivalently

O(√n) repetitions of the circuit of Fig. 3.7, each requiring O(log(n)) elementary quantum

gates), the walk will be found at the marked node with sufficiently high probability.

Figure 3.8: The probability to find the walk along the hypercube at each node againstthe number of steps.

A similar search algorithm and quantum circuit can be constructed to search for a marked

sub-cube of the hypercube, generalising the single-node search of [2]. We detail such an

algorithm in Section 3.6, and show that a marked d-cube can be found within the t-

cube using O(√

2t−d) queries to an oracle storing this marking. This proof is constructed

using a method analogous to those employed in Shenvi et al. [2] and Reitzner et al [13],

approximating the evolution of the quantum walk as a rotation in a two-dimensional plane,

oscillating between the initial state (an equal superposition of all states) and a state that

has high overlap with the marked nodes. A quantum circuit implementing such a walk

along the t-cube containing a marked d-cube is shown in Fig. 3.9.

3.5.3 Complete Graph

Similar results pertain to the complete graph on n = 2t nodes. Here we consider each node

to possess a self loop, so that each of the n nodes has degree n. Combining the circuit which

we constructed in [33], which efficiently implements a step of the walk along a complete

graph, with the search algorithm of Reitzner et al. [13] yields a complete implementation

of the search for the marked node. Fig. 3.10 presents this implementation, for a single

step of the walk. The shifting and coin operations are applied to the topology of the

complete graph exactly as with the hypercube. As in the hypercube example above, for a

complete graph on n = 2t nodes, O(t) node qubits are required. Note briefly that as the

30


...

...node

......

G G π

•... · · · · · · ...subnode • • • • •

Figure 3.9: Quantum circuit implementing a step of the quantum search along thet-dimensional hypercube, containing a marked d-cube. The circuit contains t ‘node’qubits and log(t) ‘subnode’ qubits, and the marked d-cube is represented by the states|x1, · · · , xd, 0 · · · 0〉.

complete-n graph contains n nodes, of maximum (and in this case uniform) valency n, a

lower bound of O(log(n)) node and subnode qubits are required. In this case no particular

mapping between the bit-strings and the nodes is necessary - as it is a complete graph,

for our purposes any labelling of the vertices is equivalent. As each node has degree t,

O(t) subnode qubits are also required, together with O(t) auxiliary qubits to implement

the CtG and Ctπ gates†. Hence O(t) = O(log(n)) elementary 2-qubit gates are again

sufficient to implement a step of the walk along this graph. The marked node is found

with sufficiently high probability after O(log(n)) steps (see [13]), hence in total it requires

O(√nlog(n)) = O(

√n) 2-qubit quantum gates to locate the marked node with sufficiently

high probability. Inclusion of an amplitude amplification subroutine involves an additional

O(log(n)) queries to the ‘oracle’, leaving the O(√n) complexity unchanged.

Walks along highly symmetric variants of the complete graph can also be efficiently imple-

mented. For instance consider a complete graph on 2t vertices together with an arbitrary

labelling of the vertices from 1 to 2t. Removing edges between nodes whose labels differ

by a multiple of 2 leads to a regular graph of degree 2t−1 shown in Fig. 3.11 - the complete

bipartite graph. A single step of a quantum walk along this graph can be implemented by

the circuit of Fig. 3.11, an even simpler circuit than for the complete 2t graph.

†Here the notation CtG and Ctπ follows that of generalised CNOT gates used in this work. Hencea CtG gate describes the operation of a Grover coin acting dependent on the state of t control qubits:namely a generalised controlled-G gate.

31


×• • ×

node ...... × ×

G G π

××

subnode ......

××

Figure 3.10: Quantum circuit implementing a step of the quantum search along the com-plete graph on 2t nodes. The circuit contains the minimum t ‘node’ qubits and t ‘subnode’qubits, and the marked node is represented by the state |01 · · · 00〉.

××

node ××

H ×subnode H ×

H ×

Figure 3.11: Quantum circuit implementing a quantum walk along a complete 16-graphwith every second edge removed.

3.5.4 Twisted Toroid

As an extension to the t-cube, we considered the possibility of marked node searching

along the twisted toroidal family of graphs employed in [87] to set up QC-universal toroidal

lattice cluster states. Formally, a graph in this family of dimension s× t is constructed by

taking an s× t grid, and associating the endpoints with each other as in the construction

of a simple toroid, with the modification : 1, j ↔ N, (j − 1) mod t and i, 1 ↔(i − 1) mod s, t, where ‘↔’ denotes an edge. Here i, j refers to the corresponding

point on the grid, where 1 ≤ i ≤ s, 1 ≤ j ≤ t. Fig. 3.12 shows a member of this family of

graphs, of dimension 20× 20.

This family of graphs is highly symmetric, and can be completely characterised by a small

number of parameters (growing logarithmically with the size of the graph). Given this

symmetry, it is a good candidate for a graph on which both quantum search and a direct

32


Figure 3.12: A twisted toroid of dimension 20× 20.

implementation of quantum walks are efficient. We previously showed in [33] that the

latter criteria for efficiency holds, in that for a graph of size n a step of an unbiased walk

requires O(log(n)) elementary quantum gates. Fig. 3.13 extends this result, providing a

circuit explicitly performing a step of a search for a marked node, given a twisted toroid

of arbitrary dimension 2s × 2t.

For this quantum search, the coin and shifting operator are again defined as in the hyper-

cube example above, with the coin operator applying the Grover coin to every node except

the marked node, on which the negative identity operator is applied. Each node has degree

4, and the ‘node’ states are represented by bit strings comprising the x- and y-coordinates

of the grid from which the twisted toroid was constructed, such that the node (i, j) is

represented by the bit string (i−1) concatenated with (j−1). Hence the shifting operator

applied to node (i, j) can be implemented by either incrementing or decrementing the bit

string associated with each of i and j (modulo 2s and 2t respectively), depending on the

coin state of this node. Note that the first three operations again implement the perturbed

coin operator, with the remaining set of gates implementing the shifting operator. As in

all cases considered here, the symmetry of the graph considered allows the implementation

of large sections of the shifting operator (corresponding to discrete sections of the graph)

with a single quantum gate.

Given that each increment and decrement gate can be implemented using O((s + t)2) 2-

qubit gates via the circuits of Section 3.3.1, we are now in a position to calculate an upper

bound on the number of 2-qubit gates required to implement Fig. 3.13. For a twisted

toroid of dimension 2s× 2t, with n = 2s× 2t nodes, s ‘x-coord’ and t ‘y-coord’ qubits will

33


incr decr incr decr

•...

...x-coord • • • •

• •

incr decr

•

decr incr...

...y-coord • •

G G π• • • • ×

subnode • • • •

Figure 3.13: Quantum circuit implementing a step of the quantum search along thetwisted toroid of dimension 2s × 2t. The circuit contains s ‘x-coord’ qubits and t ‘y-coord’qubits, and the marked node is represented by the state |0 · · · 10, 1 · · · 00〉, with the delimitingcomma separating the ‘x-coord’ qubits from the ‘y-coord’ qubits.

be required, together with 2 ‘subnode’ qubits (each vertex has valency 4). The controlled

G and π gates will be implementable using O(s+ t) 2-qubit gates, and since the increment

and decrement gates on i qubits require O(i2) 2-qubit gates, the controlled increment and

decrement operators are implementable using O(s2t + t2s) 2-qubit gates (note that since

the degree of the graph is fixed at 4, only 2 subnode qubits will be required regardless of

s and t). Hence a step of the quantum walk along a twisted toroid on O(n) nodes can be

implemented using O(log(n)) 2-qubit gates.

It remains to show whether or not the circuit of Fig. 3.13 can perform an efficient search

for the marked node - in other words, whether for a twisted toroid of dimension 2s×2t, the

marked node can be found with O(√

2s ×√

2t) iterations of this circuit. To examine this,

the behaviour of the walk was analysed for a range of different toroid sizes. Regardless

of the size, the search behaviour was in one respect successful, in that the probability to

find the walk at the marked node reached a sufficiently high level after a certain number

of steps, with fixed periodic behaviour. The approximate number of steps required was

found for a range of twisted toroid sizes by calculating the period of the success probability

(to find the walk at the marked node) with number of steps. The resulting correlation

between toroid size (in number of vertices) and approximate number of steps required to

reach maximum success probability is shown in Fig. 3.14. These numerical results show

that for a twisted toroid with n nodes, O(√n) steps of the walk are sufficient to find

the marked node (accounting for the addition of an amplitude amplification subroutine),

matching, up to a factor of log(n), the theoretical lower bound for unstructured search [80].

34


0 200 400 600 800

0

10

20

30

40

50

Size of graphSt

eps

requ

ired

Figure 3.14: The period of the probability function, showing the number of steps requiredfor the quantum walk search against the size of the twisted toroid.

Putting these results together, we see that this family of twisted toroids satisfy both criteria

for efficient search via quantum walks, yielding for a graph on n nodes both an efficient

implementation of the walks themselves (using O(log(n)) 2-qubit gates), and requiring

O(√n) elementary 2-qubit gates to find a marked node with arbitrarily high probability.

3.5.5 Glued Trees

Given the results of Childs et al. [10], in which quantum walks are shown to traverse a

family of ‘glued trees’ exponentially faster than any possible classical algorithm, relative

to a quantum oracle, we include a discussion of quantum walks along glued trees in the

non-oracular setting. Note that the algorithm presented in [10] employs continuous-time

quantum walks, while in Cleve et al. [88] it was shown to also be implementable by discrete-

time quantum walks. Both require the use of a quantum oracle. In both cases, the ‘glued

trees’ in question are a family of graphs, each comprising two balanced binary trees of

depth t with random interconnections between the central level, as in Fig. 3.15(a), such

that every node except for the left- and right-most nodes (termed the entrance and exit

nodes respectively) have a valency of 3. Given a quantum walker initialised in the entrance

node state, the problem is to find the exit node efficiently. The key component to this

problem is that a labelling for the nodes is not a priori known. Instead an oracle is

provided which accepts bit-strings as input, outputting the neighbouring nodes in the

case where this bit-string corresponds to the labelling of a node†. In [10] it is proven

that no classical algorithm can solve this problem with sufficiently high probability in

†Hence the labelling of the nodes is either explicitly or implicitly encoded into the operation of theoracle.

35


polynomial time (relative to the tree depth t).

Figure 3.15: Binary glued trees with random (a) and regular (b) interconnections betweenthe central levels.

In the non-oracular case, efficient implementation of a quantum walk along the glued

trees is not possible given random interconnections between the central levels (as in Fig.

3.15(a)), since this would be equivalent to performing a random permutation of 2t states

in time O(poly(t)). In other words, in the absence of some regular labelling of the nodes,

the underlying structure of the graph cannot be exploited to efficiently encode the shifting

operator into O(logt) qubits. Instead we are restricted to considering regular interconnec-

tions, such as those of Fig. 3.15(b). Here ‘regular’ interconnections are explicitly those

that can be completely characterised by O(poly(t)) parameters. The algorithm of Childs

et al. [10] requires a symmetric coin operator - hence we use the Grover coin, defined on d

dimensions by (Gd)i,j = 2d − δi,j , the only purely real non-trivial symmetric coin [19] (up

to a factor of ±1). We also restrict the shifting operator to S2 = I, where I is the identity

operator. In this case an efficient quantum circuit can be constructed, for example that

36


of Fig. 3.16, for tree depth 4 (with 62 nodes).

RL RR

node

×

HG3

×subnode • • • • • •

•incr

decr

•level • •

side of tree • • •

|1〉 • • •

Figure 3.16: Quantum circuit implementing a quantum walk along a glued tree with aregular labelling of the nodes. Here the G3 operator acts on only 3 of the 4 subnode qubits,leaving the state |10〉 unaffected.

Here the G3 gate represents a three dimensional Grover coin operator acting on two qubits

(mixing three of the four states, while the fourth is not accessed). Following the complexity

analysis of the preceding sections, for a tree depth of t, the circuit requires t + log2t + 5

qubits, together with O(t2) elementary 2-qubit gates.

37

3.6. SEARCH FOR A MARKED SUBCUBE OF A HYPERCUBE

3.6 Search for a Marked Subcube of a Hypercube

Quantum walk based search algorithms began with the work of Shenvi et al. [2], estab-

lishing that O(√N) steps of a quantum walk are sufficient to find a single marked node

of a hypercube with N vertices. The algorithm they construct employs an oracle, which

at each step of the quantum walk performs the function

f(x) =

1 if x = xtarget

0 otherwise(3.1)

where xtarget represents the marked node. They prove that O(√N) queries to this oracle

suffice by showing that the walk evolution operator approximately implements a rotation

between the initial state and a state in which the marked node has a high probability

amplitude.

The results of [2] represented a significant advance in quantum search algorithms, leading

to a series of subsequent papers [9, 13, 78, 89–94]. Several of these papers deal specifically

with the methods and situation described in [2], considering various optimisations and

extensions to this hypercube search problem [78,89,90,92,93]. Indeed, these results remain

highly relevant, continuing to stimulate new work.

In this section we generalise the results of [2] to the case where a sub-cube of the hypercube

is marked, and show that O(√N) queries to the oracle also suffice to find such a marked

subgraph. Specifically, given an n-cube with N = 2n vertices, we show that a marked

d-cube within this n-cube (where d is sufficiently less than n) can be found using O(√N)

queries to a particular oracle. The situation where a single vertex is marked occurs in

the extremal case where d = 0. While such a result is to be expected, it nevertheless

represents a neat generalisation to the methods of [2].

3.6.1 Walk on an Unmarked Hypercube

We consider a walk on the n-cube G, containing N = 2n vertices, each labelled by an n-

length bit-string such that two vertices ~x and ~y are connected by an edge if their Hamming

distance is 1, in other words if |~x−~y| = 1. Each of the 2n nodes has valency n, such that the

Hilbert space of the walk is given by H = H2n ⊗Hn, where each state of H corresponds

38


to a di-edge of G†. We will follow a slight modification of the quantum walk notation

discussed in Chapter 2, such that a state of the walk corresponding to vertex ~x is given

by |~x, c〉, where 1 ≤ c ≤ n corresponds to the edge incident with ~x that is associated with

the cth bit in ~x, or alternatively the cth edge-dimension of G.

In this basis, the shifting operator is given by

S =∑

~x

n∑

c=1

|~x⊕ ~ec, c〉〈~x, c| , (3.2)

where ⊕ represents bit-wise addition modulo 2 and ~ec is the cth basis vector on the hyper-

cube. As G is unmarked, the coin operator is unbiased, and the Grover coin will be used,

such that

C = I ⊗G, (3.3)

where G is the Grover coin on n dimensions defined in Eq. (2.5).

This walk on the hypercube was studied extensively in Moore and Russell [77], in which

they show that the non-trivial eigenvalues and eigenvectors of U = S · C are

v±k = e±iωk = 1− 2k

n± 2i

n

√k(n− k) , (3.4)

|v±~k 〉 =∑

~x

n∑

c=1

(−1)~k·~x 2−n/2√

2|~x, c〉 ×

1/√k if kc = 1

∓i/√n− k if kc = 0

, (3.5)

where k = |~k| and kc is the cth component of ~k. For further discussion regarding this

‘unmarked’ eigenspectrum, see, for example, [2, 77, 93]. We briefly note that the equal

superposition over all states, given by

|ψ0〉 =1√2nn

∑

~x

n∑

c=1

|~x, c〉 , (3.6)

is trivially an eigenvector of U with eigenvalue 1, since C is unbiased and G is a regular

graph. Also note that the eigenvalues and eigenvectors of U are in complex conjugate

pairs, as U is a real-valued matrix. This also holds for the perturbed evolution operator

†Note that to retain consistency with the notation of previous chapters, we order components of theHilbert space by position, then coin state respectively. This is a reversal of the ordering used in [2].

39


U ′ described below.

3.6.2 The Marked Hypercube

We consider a marking of a d-dimensional sub-cube of G. Without loss of generality, we

take this sub-cube H ⊂ G to be labelled by (i1, . . . , id, 0, . . . , 0), comprising the vertices in

which the last n − d bits are 0. Relative to this marking, the coin operator is perturbed

such that C~x = −I for all ~x ∈ H, forming the perturbed coin operator

C ′ = I ⊗G−∑

~x∈H|~x〉〈~x| ⊗ (I +G). (3.7)

Substituting the form of the Grover coin in Eq. (2.5), the corresponding perturbed evolu-

tion operator becomes

U ′ = S · C ′ (3.8)

= U − 2S · (∑

~x∈H|~x〉〈~x| ⊗

∣∣sC⟩ ⟨sC∣∣), (3.9)

where∣∣sC⟩

= 1√n

∑nc=1 |c〉, the equal superposition over all n edges.

3.6.3 Collapsing to a Walk on the Line

As in [2], we analyse the evolution of this perturbed walk by first reducing the dimension

of the Hilbert space, simplifying the analysis of the eigenspace of the walk. Rather than

redefining the basis states according to their Hamming weight as in [2], the perturbation

of the evolution operator considered above leads naturally to a set of basis states defined

relative to their distance from the subgraph H.

We note briefly that this mention of a ‘natural’ reduction in the Hilbert space dimension

is simply a reflection of the symmetries present in the unitary evolution operator U ′.

The hypercube begins with an automorphism group corresponding to the hyperoctahedral

group, and the original unbiased evolution operator U preserves this automorphism group,

in that for all φ ∈ Aut(G), U commutes with the operator implementing the permutation

φ. Marking the d-cube H within G effectively corresponds to a set-wise stabilisation of

Aut(G) relative to H, with the result that U ′ now commutes only with the subset of

Aut(G) remaining after this set-wise stabilisation.

40


As this subset of Aut(G) comprises the elements of Aut(G) with fixed Hamming distance

from H, the related set of basis states of H is in this sense the ‘natural’ reduction relative

to the perturbation.

Hence we define a collapsed basis consisting of 3(n − d) + 1 basis states,

|0,M〉 , . . . , |n− d,M〉, |1, L〉 , . . . , |n− d, L〉, |0, R〉 , . . . , |n− d− 1, R〉, where

|x,M〉 =

√1

2d d(n−dx

)∑

d(~x)=x

∑

i≤d|~x, i〉 , (3.10)

|x, L〉 =

√1

2d x(n−dx

)∑

d(~x)=x

∑

i∈A|~x, i〉 , (3.11)

|x,R〉 =

√1

2d (n− d− x)(n−dx

)∑

d(~x)=x

∑

i∈A

|~x, i〉 , (3.12)

where A = j > d : ~xj = 1, A = j > d : ~xj = 0 and x := d(~x) is the distance of ~x from

H, explicitly x =∑

i>d xi. The value (n−d−x) is also denoted by x, where 0 ≤ x ≤ n−d,

such that, for example, any state (i1, . . . , id, 1, . . . , 1) has an x value of 0.

Here the L and R states move towards and away from H respectively, and the M states

remain at the same distance from H, under the application of S.

We note a few properties of the perturbed walk in this collapsed basis. The shifting

operator in this basis is rewritten as

S =n−d∑

x=0

|x,M〉〈x,M |+n−d−1∑

x=0

(|x,R〉〈x+ 1, L|+ |x+ 1, L〉〈x,R|), (3.13)

and the unperturbed coin operator becomes

C = |x〉〈x| ⊗

φ(d2)− 1 φ(xd) φ(xd)

φ(xd) φ(x2)− 1 φ(xx)

φ(xd) φ(xx) φ(x2)− 1

= |x〉〈x| ⊗

2dn − 1 2

√xdn

2√xdn

2√xdn

2xn − 1 2

√xxn

2√xdn

2√xxn

2xn − 1

, (3.14)

where φ(i) := 2√i

n . Here the first part of C acts on the position space 0, . . . , n− d and the

second part is expressed in the basis |M〉 , |L〉 , |R〉, so that in the limit where d = 0, the

resulting 2 × 2 matrix representation is obtained by removing the first row and column

41


from the representation above.

Combining the shifting and coin operators of Eqs. (3.13) and (3.14) yields the walk evo-

lution operator

U = S · C

=n−d−1∑

x=0

|x,R〉 (C2,2 〈x+ 1, L| + C2,3 〈x+ 1, R| + C2,1 〈x+ 1,M | )

+

n−d∑

x=1

|x, L〉 (C3,2 〈x− 1, L| + C3,3 〈x− 1, R| + C3,1 〈x− 1,M | ) (3.15)

+n−d∑

x=0

|x,M〉 (C1,2 〈x, L| + C1,3 〈x,R| + C1,1 〈x,M | ).

The perturbed coin operator is defined by C ′ = C + ∆C, where:

∆C = |0〉〈0| ⊗

−φ(d2) −φ(d(n− d))

−φ(d(n− d)) −φ((n− d)2)

= |0〉〈0| ⊗ − 2

n

d

√d(n− d)

√d(n− d) (n− d)

, (3.16)

where the first part acts on the position |0〉, and the second part on the space spanned

by |M〉 , |R〉. This perturbation of the coin operator can be explicitly expressed in the

basis state form as

∆C = − 2

n(√d |0,M〉 +

√n− d |0, R〉 )(

√d 〈0,M | +

√n− d 〈0, R| ). (3.17)

Combining Eqs. (3.13) and (3.17) yields the perturbed walk evolution operator U ′ =

U + ∆U , such that

∆U = − 2

n(√d |0,M〉 +

√n− d |1, L〉 )(

√d 〈0,M | +

√n− d 〈0, R| ). (3.18)

Note that in the reduced state, the form of |ψ0〉 is

|ψ0〉 =

√2d

2nn

n−d∑

x=0

√(n− dx

)(√d |x,M〉+

√x |x, L〉+

√x |x,R〉), (3.19)

42


contrasted to its form relative to the original basis:

|ψ0〉 =1√2nn

∑

~x,d

|~x, d〉 . (3.20)

The eigenvectors |v±~k 〉 of the unperturbed walk given in Eq. (3.5) can be combined to form

eigenvectors |ωk〉 of U in the collapsed space, such that

|ωk〉 =1√(n−dk

)∑

d(~k)=k

|v±~k 〉. (3.21)

Recall that the eigenvalues of unperturbed walk (in collapsed basis) are split into two

types, trivial and non-trivial (as in Moore and Russell [77]). Trivial eigenvalues are simply

+1 and −1 with multiplicity 1 and n − d respectively. The non-trivial eigenvalues take

the form

e±iωk = 1− 2k

n± 2i

n

√k(n− k), (3.22)

where k = 1, . . . , n− d.

3.6.4 Search Algorithm

Given this perturbed quantum walk on the hypercube, collapsed to a walk along the line

according to the inherent symmetry of U ′, we wish to construct a search algorithm and

show that it finds the marked sub-cube after O(√

2n) applications of U ′. The basic form

of the algorithm and the proof of its running time mirrors the algorithm described in [2],

and is described as follows.

1. The walk is initialised in state |ψ0〉, the equal superposition of all states in the

original basis.

2. An oracle is provided that performs the function of Eq. (3.1), returning 1 if the walk

is at a marked node and 0 otherwise. This oracle is used to perform the perturbation

to the coin operator, ∆C.

3. Apply the perturbed evolution operator, U ′, t =√

nn−d

√2n−d−1 π

2 times.

4. Measure the state in the |~x, d〉 basis. If the resulting state is not marked, apply the

shifting operator to this state and check again.

43


5. If a marked state is found, check whether each of the neighbouring states in turn are

marked.

We show that at the conclusion of this algorithm, all nodes of the marked sub-cube have

been identified, with probability 1−O( 1n−d).

As a brief note on the values of n, d under consideration, we will assume that n−d = O(n),

such that the proportion of nodes that are marked remains low, with 2d

2n = O(e−n).

The method of proof will follow that of Shenvi et al. [2]. We construct two approximate

eigenvectors of U ′; the initial state |ψ0〉 and a second state, |ψ1〉, having high overlap

with the marked subgraph H. We show that the evolution of (U ′)t |ψ0〉 can be well-

approximated to lie within the space spanned by two eigenvalues | ± ω′0〉 of U ′, each of

which is in turn well-approximated by linear combinations of |ψ0〉 and |ψ1〉. As a result, the

evolution of (U ′)t |ψ0〉 can be approximately modelled as a sinusoid, oscillating between the

initial state |ψ0〉 and the state |ψ1〉, a close approximation to the target subgraph H. Each

step of the walk is shown to correspond to a rotation of approximately√n− d/

√2n−d−1 n,

hence for n− d = O(n), O(√

2n−d) = O(√

2n) calls to the oracle are sufficient to complete

the search algorithm.

3.6.5 Proof of Running Time

Before considering the eigenvectors relevant to the evolution of (U ′)t |ψ0〉, we will analyse

the eigenvalues of U ′. As in the case where d = 0, investigated in [2], we first show that

for general d there are exactly two eigenvalues of U ′ with real part greater than 1− 2/3n.

Theorem 3.6.1. There are at most two eigenvalues of U ′ with real part greater than

1− 2/3n.

Proof. Assume that three such eigenvalues exist, labelled by eiω′j , j ∈ 0, 1, 2, with cor-

responding eigenvectors |ω′j〉. Then

Re2∑

j=0

〈ω′j |U ′|ω′j〉 = Re2∑

j=0

eiω′j 〈ω′j |ω′j〉

> 3− 2/n. (3.23)

44


In considering the Re 〈α2|∆U |α2〉 term, first note that 〈α2|ψ−〉 = 0. Introducing the

state† |ψ+〉 = 1√2n

(√n− d |0, R〉+

√n− d |1, L〉), we can expand ∆U , given in Eq. (3.18),

in terms of |ψ−〉 and |ψ+〉 as

∆U = (|ψ−〉 − |ψ+〉)(|ψ−〉+ |ψ+〉) (3.31)

= |ψ−〉〈ψ−|+ |ψ−〉〈ψ+| − |ψ+〉〈ψ+| − |ψ+〉〈ψ−| .

Then since 〈α2|ψ−〉 = 0 from Eq. (3.26), we have

〈α2|∆U |α2〉 = −|〈ψ+|α2〉|2 ≤ 0. (3.32)

Finally, note that Re 〈αi|U ′|αi〉 ≤ 1 for all |αi〉, hence

Re

2∑

j=0

〈αj |U ′|αj〉 ≤ 2 + Re 〈α2|U ′|α2〉. (3.33)

Combining Eqs. (3.33), (3.30) and (3.32) yields

Re2∑

j=0

〈αj |U ′|αj〉 ≤ 3− 2/n. (3.34)

However this contradicts Eq. (3.28), and hence the original assumption is invalid, and

there are at most two eigenvalues of U ′ with real part greater than 1− 2/n.

Consider the initial state |ψ0〉, defined relative to the collapsed basis in Eq. (3.19). Sub-

stituting Eq. (3.18), we see that

∆U |ψ0〉 = − 2

n(√n− d |1, L〉+

√d |0,M〉)(n

√2d√

2nn), and (3.35)

〈ψ0|∆U = − 2

n(√n− d 〈0, R|+

√d 〈0,M |)(n

√2d√

2nn), (3.36)

and so, since 〈ψ0|U |ψ0〉 = 1, we have

〈ψ0|U ′|ψ0〉 = 1− 2 · 2d2nn

(n− d+ d)

= 1− 1

2n−d−1. (3.37)

†Note that in the extremal case where d = 0, the M terms vanish and the |ψ±〉 vectors defined here inthe case of generalised d match those of [2].

46


Hence in the limiting case where (n − d) is large, |ψ0〉 is approximately an eigenvector

of U ′, with eigenvalue 1. Note that in the extremal case where d = 0, this simplifies to

1− 1/2n−1, matching the results of [2].

As in [2], we wish to find a second approximate eigenvector, labelled by |ψ1〉. A suitable

form is

|ψ1〉 =

n−d2−1∑

x=0

1√2(n−d−1x

) |x,R〉 −1√

2(n−d−1x

) |x+ 1, L〉

1

c, (3.38)

where c is a normalisation factor,

c =

√√√√√n−d2−1∑

x=0

1(n−d−1x

) . (3.39)

Substituting Eqs. (3.15) and (3.18), we obtain

U ′ |ψ1〉 = |ψ1〉−√n− d

c n√

2(n−d−1(n−d)/2

)(√

d∣∣n−d

2 ,M⟩

+√n− d

∣∣n−d2 + 1, L

⟩+√n− d

∣∣n−d2 − 1, R

⟩), (3.40)

and hence

〈ψ1|U ′|ψ1〉 = 1− (n− d)(2n− d)

2c2n2(n−d−1(n−d)/2

) . (3.41)

Now as in [2], 1 < c2 < 1 + 2/(n− d) for sufficiently large n− d, hence apart from a small

residue, |ψ1〉 is approximately an eigenvector of U ′ with eigenvalue 1.

Having established these two approximate eigenvectors of U ′, we are now in a position to

prove the following eigenspectrum property.

Theorem 3.6.2. There are exactly two eigenvalues of U ′ with real part greater than

1− 2/3n.

Proof. Assume U ′ has no such eigenvalues. Then cos ω′j < 1− 2/3n for all j. Hence from

47


Eq. (3.37) we have

1− 1/2n−d−1 = Re 〈ψ0|U ′|ψ0〉

=∑

j

|〈ψ0|ω′j〉|2 cos ω′j

< (1− 2/3n)∑

j

|〈ψ0|ω′j〉|2 (3.42)

= 1− 2/3n.

For large n− d this leads to a contradiction, hence U ′ has at least one such eigenvalue.

Similarly, if we assume U ′ has exactly one such eigenvalue, eiω′0 , we have

1− 1/2n−d−1 = Re 〈ψ0|U ′|ψ0〉

= |〈ψ0|ω′0〉|2 cos ω′0 +∑

j 6=0

|〈ψ0|ω′j〉|2 cos ω′j (3.43)

≤ |〈ψ0|ω′0〉|2 +(1− |〈ψ0|ω′0〉|2

)(1− 2/3n) ,

which implies that

|〈ψ0|ω′0〉|2 ≥ 1− 3n

2n−d. (3.44)

Similarly, applying the same process using |ψ1〉 and Eq. (3.41) yields

|〈ψ0|ω′0〉|2 ≥ 1− 3(n− d)(2n− d)

4c2n(n−d−1(n−d)/2

) , (3.45)

and so (as in [2], Theorem 3) we have a contradiction. That is, the orthonormality of |ψ0〉and |ψ1〉 together with Eqs. (3.44) and (3.45) implies that

1 = 〈ω′0|ω′0〉

≥ |〈ψ0|ω′0〉|2 + |〈ψ1|ω′0〉|2 (3.46)

≥ 2− 3n2n−d

− 3(n− d)(2n− d)

4c2n(n−d−1(n−d)/2

) ,

which is not true for large n− d. Hence U ′ has exactly two such eigenvalues.

Although a sufficiently large value of n − d is assumed in the above proof, we note that

a numerical analysis indicates that exactly two such eigenvalues exist for all values of n

48


0 20 40 60 800.92

0.94

0.96

0.98

1.00

n-d

Figure 3.17: The real component of the three largest eigenvalues of U ′ plotted againstn− d. Here n is fixed at 80. The blue solid line shows 1− 2

3n , and the remaining solid lines

are at 1− 2n and 1− 3

n respectively. Note that as n− d→∞, Re eiω′j → 1− j+1

n .

and d except where d = n− 1. Furthermore, the real part of these eigenvalues e±i ω′j of U ′

limits to the value of 1 − (j + 1)/n from below for large n − d, as shown in Fig. 3.17 for

n = 80.

As noted previously, these eigenvalues are complex conjugate pairs, and hence will be

denoted e±iω′0 , with corresponding eigenvectors being |ω′0〉 and |−ω′0〉 = |ω′0〉∗. The next

step is to show that these eigenvectors can be well-approximated by the linear combinations

of the approximate eigenvectors |ψ0〉 and |ψ1〉.

Theorem 3.6.3.

|ω′0〉 =√p0 |ψ0〉+

√p1e

iη |ψ1〉+√

1− p0 − p1 |r0〉

|−ω′0〉 =√p0 |ψ0〉+

√p1e−iη |ψ1〉+

√1− p0 − p1 |r0〉∗ , (3.47)

where p0 = |〈ψ0|ω′0〉|2, p1 = |〈ψ1|ω′0〉|2 and |r0〉 is a normalised vector orthogonal to |ψ0〉and |ψ1〉, such that p0, p1 ≈ 1/2 and eiη ≈ i for large n− d.

Proof. The proof proceeds as in the corresponding proof in [2] (Theorem 4), with the only

variance due to the differing forms of Eqs. (3.37) and (3.41). Specifically, as | ± ω′0〉 are

complex conjugates, and |ψi〉, i = 1, 2 are real vectors, we have |〈ψi| ± ω′0〉|2 ≤ 1/2,

i ∈ 1, 2, and hence p0, p1 ≤ 1/2. Then Eq. (3.37) yields

p0 ≥ 1/2− 3n

2n−d+1(3.48)

49


and Eq. (3.41) yields

p1 ≥ 1/2− 3(n− d)(2n− d)

8c2n(n−d−1(n−d)/2

) . (3.49)

Now |ω′0〉 can be written up to a global phase as

|ω′0〉 = |〈ψ0|ω′0〉| |ψ0〉 + |〈ψ1|ω′0〉| eiη |ψ1〉 +√

1− p0 − p1 |r0〉 , (3.50)

and so the two expressions of Eq. (3.47) follow. To approximately determine the value

of η we follow the same general method as in [2] (Theorem 4). As |ω′0〉 and | − ω′0〉 are

orthogonal, we have

0 = 〈−ω′0|ω′0〉

= p0 + p1

(eiη)

+ (1− p0 − p1) Re 〈r∗0|r0〉, (3.51)

yielding

Re(eiη)2

=−p0 − (1− p0 − p1) Re 〈r∗0|r0〉

p1. (3.52)

Then assuming Re 〈r∗0|r0〉 ≥ 0 we have

Re(eiη)2 ≤ −p0

p1≤ −2p0 = −1 +

3n

2n−d, (3.53)

together with (substituting Re 〈r∗0|r0〉 = 1, and using the identity 1−xx ≈ 1 + 4(1

2 − x) for

x ≈ 12)

Re(eiη)2 ≥ −p0 − (1− p0 − p1)

p1

=p1 − 1

p1

≈ −1− 4

(3(n− d)(2n− d)

8c2n(n−d−1(n−d)/2

)). (3.54)

Hence −1 − ε1 ≤ Re(eiη)2 ≤ −1 + ε2, where ε1 and ε2 are positive values, each approx-

imately zero for large n − d. Hence eiη ≈ i. The same arguments apply, with ε1 and ε2

reversed, if Re 〈r∗0|r0〉 ≤ 0.

By rearranging Eq. (3.47) we can now express the initial state |ψ0〉 as a linear combination

50


of the eigenvectors |±ω′0〉, together with a small components attributed to a residual vector

|r〉, orthogonal to each of the |±ω′0〉 eigenvectors, such that

|ψ0〉 =√p0(|ω′0〉+ |−ω′0〉) + δ |r〉 , (3.55)

where δ =√

1− 2p0 = O(√n/2n−d).

Hence |ψ0〉 ≈ 1√2(|ω′0〉+ |−ω′0〉), and applying U ′ t times we arrive at

(U ′)t |ψ0〉 ≈√p0(eiω

′0t|ω′0〉+ e−iω

′0t|−ω′0〉)

≈ 1√2(cos ω′0t |ψ0〉 − sin ω′0t |ψ1〉 , (3.56)

and all that remains is to determine the value of the angle ω′0.

Theorem 3.6.4. sin ω′0 ≈ −√n−d

c√

2n−d−1nfor large n− d.

Proof. Firstly, note that from Eq. (3.47) we have eiω′0 = 〈ω′0|U ′|ω′0〉 ≈ 〈α|U ′|α〉 for large

n− d. Now

〈α|U ′|α〉 = 12(〈ψ0|+ i 〈ψ1|)(U ′ |ψ0〉+ i U ′ |ψ1〉)

= 12(〈ψ0|U ′|ψ0〉 − 〈ψ1|U ′|ψ1〉+ i 〈ψ0|U ′|ψ1〉+ i 〈ψ1|U ′|ψ0〉), (3.57)

and so

sin ω′0 ≈ 〈ψ0|U ′|ψ1〉+ 〈ψ1|U ′|ψ0〉. (3.58)

Referring to Eqs. (3.35) and (3.40), and noting that |ψ0〉 and |ψ1〉 are orthonormal, we

obtain

sin ω′0 ≈ −√n− d

c√

2n−d−1n. (3.59)

Now note that c ≈ 1 for large n− d, so in the case where n− d = O(n), after O(√

2n−d−1)

steps the walk will be approximately in state |ψ1〉 with high probability. More explicitly,

for large n− d, after t =√

nn−d

√2n−d−1 π

2 steps of the walk, we have

(U ′)t |ψ0〉 ≈ − |ψ1〉 . (3.60)

51

3.7. CONCLUSIONS AND OPEN QUESTIONS

0 20 40 60 80 100 120 1400.75

0.80

0.85

0.90

0.95

1.00

n-d

Prob

abili

ty

Figure 3.18: The probability to measure |ψ1〉 at either |0, R〉 or |1, L〉 (solid line) togetherwith a plot of 1− 1

n−d−2 (dotted line).

For large n− d the state |ψ1〉 in turn has the form

|ψ1〉 ≈ 1√2(|0, R〉 − |1, L〉). (3.61)

In fact, the probability to measure the state |ψ1〉 at either |0, R〉 or |1, L〉 is fixed for a

given value of n− d, and this probability is plotted against increasing n− d in Fig. 3.18.

Running through the algorithm described in Section 3.6.4 once more, we consider the case

where n − d is large, and initialise the walk in the state ψ0†. After t =

√nn−d

√2n−d−1 π

2

steps of the walk, the state of the system is measured in the original, uncollapsed basis.

With probability 1/2 − O( 1n−d), the measured state will be at a marked node. Else the

shifting operator S is applied to the measured state, and with probability 1/2 − O( 1n−d)

the resultant state is at a marked node. At this stage a single marked node has been

found with probability 1−O( 1n−d). We then perform n further calls to the oracle, testing

sequentially whether each of the n neighbouring nodes are marked. In the process the

d marked edge-dimensions are found, revealing the entire marked subgraph, requiring in

total O(√

nn−d

√2n−d)) calls to the oracle.

3.7 Conclusions and Open Questions

We have presented here several families of highly symmetric graphs all amenable to both ef-

ficient quantum search and exact, efficient quantum circuits implementing quantum walks

along them. The examples given are intended to provide a ‘proof of concept’; extensions to

†As ψ0 is a product state, it can be efficiently prepared.

52


more complex variations can also be made using the concepts and methods described here.

For example, both composites of the provided graphs and graphs involving a small number

of perturbations or imperfections are also amenable to efficient circuit implementations.

By explicitly characterising the internal action of the oracle, these circuits provide a prac-

tically implementable method to experimentally test these quantum walk based search

algorithms.

Open questions relating to this work include:

• Can any graphs which are not highly symmetric in the sense of Section 3.1 be

efficiently implemented by quantum walks? No such examples have been found to

date, however neither has any proof eliminating this possibility been established.

• Similarly, for quantum walk searching to obtain the O(√n) lower bound, must the

underlying graph be highly symmetric in this same sense? Again, no counterexam-

ples have yet been found.

• Regarding the issue of sparse graphs, does the property of possessing efficiently

computable neighbours also imply a non-trivial automorphism group?

Further work includes extending the methods of analytical proof used in Section 3.6 to

other families of highly symmetric graphs, in particular the twisted toroid.

53

Chapter 4

Quantum Walk Based Graph

Isomorphism Algorithm

Based on B. L. Douglas and J. B. Wang’s original publication in J.

Phys. A: Math. Theor. 41 (2008) 075303. Revised content includes an

extended conclusion and discussion of subsequent work, continued in

Part II.

4.1 Introduction

The question of determining whether two given structures (for instance, algebraic or com-

binatorial) are isomorphic has been a long-standing open problem in mathematics. A

range of structures can be efficiently (i.e. in polynomial time) encoded by graphs [95],

for which it is sufficient to solve the isomorphism problem for graphs. Considerable and

continuing effort has been devoted to the graph isomorphism (GI) problem, due to both

the variety of practical applications, and its relationship to questions of computational

complexity.

Efficient GI algorithms do exist for certain restricted classes of graphs, such as trees [96],

planar graphs [97], and graphs with bounded valence [98]. Indeed, for many practical

applications, the GI problem can be viewed as ‘easy’, in that algorithms exist, such as

Brendan Mackay’s ‘Nauty’ algorithm [99], which can distinguish pairs of graphs taken

from many classes of graphs efficiently. However as yet, there is no known algorithm with

polynomial bounds to solve GI for general graphs. Currently, the best known general

GI algorithms scale with O(e√n logn) [22, 100, 101]. In fact, the exact complexity status

54

CHAPTER 4. QUANTUM WALK BASED GRAPH ISOMORPHISM ALGORITHM

of GI is unknown and generates much of the interest surrounding the problem. No NP-

completeness proof has been found, and indeed it seems likely that the GI problem is

not NP-complete, in part because the corresponding testing and counting problems are

polynomial-time equivalent, unlike the apparent case for all other known NP-complete

problems. In these cases, the counting problem seems to be much harder, although this

cannot be proven without solving the P = NP problem.

Just as isomorphism testing of many structures is equivalent to graph isomorphism, there

are various other problems with equivalent difficulty. Problems such as finding the au-

tomorphism group of a graph [102], and several subsets of GI, such as isomorphism of

regular graphs, connected graphs, undirected graphs, and graphs of diameter 2 and girth

4 are all examples of problems equivalent to GI in general [23]. In other words, solving GI

for any of the above classes of graphs would provide an efficient method for solving GI in

general.

In this work, the emphasis is on polynomial versus non-polynomial time computability

of GI. Although we detail the specific scaling of the proposed algorithms to follow, we

recognise that their current implementation may not be optimised, and it is likely that

variations based on the same general theme could be developed that scale with lower order

polynomials. For now, however, the important theoretical direction is towards establishing

a polynomially scaling GI algorithm.

In this chapter we present an algorithm utilising the action of quantum walks on graphs

to distinguish given pairs of graphs. Firstly, we introduce the concept of a quantum

walk, and relate it to the more familiar classical random walk. We then discuss possible

efficient, classical implementations of quantum walks on general graphs. A detailed dis-

cussion of a resulting GI algorithm follows, including scaling arguments, and results from

testing against various sets of graphs. Finally, the corresponding quantum GI algorithm

is discussed.

4.2 Quantum Walks

There has been considerable interest in quantum walks recently, with a variety of papers

published regarding their properties [41, 103], uses [2, 43], and possible physical imple-

mentations [81]. They are motivated by notions of quantum systems, and indeed can be

thought of as the quantum analogue of simple classical random walks.

55

4.2. QUANTUM WALKS

Consider a classical random walk on a line. In its simplest form, it consists of flipping a

coin, moving either left or right depending on the outcome of the coin flip, then repeating

these steps. The coin provides a means of randomising the chosen direction of propagation.

For example, consider an unbiased discrete walk on the integers, in which direct walking is

allowed only between consecutive integers. If the walk is started at the origin, then after

one step the walker would be at −1 or 1 with equal probability, and after two steps, at 0

with probability 0.5, and −2 or 2 with probability 0.25.

For our purposes, we identify three important differences in the definition of a quantum

walk. Firstly, since it is defined in terms of a physically implementable quantum system,

both the coin and shifting operations are constrained to be unitary. In addition, all

possible paths of the walk are sampled simultaneously - in other words, we can think of

the evolution of the walk as being the evolution of the probability amplitude distribution,

rather than involving a walk along a distinct path. Finally, as a result of the previous two

requirements, quantum walks have memory, in that the outcome of the ‘coin flip’ depends

on outcome of previous coin flips.

In fact, quantum walks in their present form have a one-step memory. This is achieved

by increasing the state space of the walk, splitting each node of the position space into

a group of d sub-nodes (or ‘coin positions’), as illustrated in Fig. 4.1, where d is the

number of outgoing ‘edges’ from the node. In this case we are assuming that the walk is

associated with some geometric structure containing edges, in which walking occurs only

along these edges. The coin operator acts differently on each sub-node, thus giving the

walk its memory.

Figure 4.1: A sample graph of six nodes is shown, split into sub-nodes. Each edge isassociated with two sub-nodes.

56


Much of the interest in quantum walks has been based around their markedly different

spreading characteristics, compared to those from simple classical random walks. Un-

biased quantum walks along a line result in spreading with visible interference patterns

such as that of Fig. 4.2 (solid line), whereas simple classical random walks give rise to a

Gaussian distribution (dashed line in Fig. 4.2). To investigate what causes these differ-

ences in propagation, we considered distributions resulting from altered quantum walks,

in which one or more of the three requirements above were relaxed. We found that both

the memory of the walk and the simultaneous sampling of all paths are needed to pro-

duce similar probability distributions to Fig. 4.2 (solid line). The combination of these

two properties results in interference/interactions between different paths of the walk, not

seen in classical random walks and certainly not possible with Markovian processes. This

interaction between paths is the key difference between classical random walks and quan-

tum walks. The third requirement, that of unitarity, is not strictly necessary to produce

such probability distributions, and can be relaxed for classical implementations of these

walks. It is however, necessary for a closed quantum system, and is an implicit property

of any quantum implementation.

Figure 4.2: Probability distributions for unbiased quantum (solid) and classical (dashed)walks along a line.

For the remainder of this chapter we will only be utilising the probability amplitude dis-

tributions associated with these walks. This removes any random aspect to both classical

and quantum walks, as the evolution of the probability distributions is in both cases a

deterministic process. It also removes one of the major nominal differences between the

definitions of quantum and classical random walks described above, with the classical

random walk now also involving sampling of all possible paths simultaneously. However,

though simultaneous, the sampling of different paths proceeds independently, unlike the

57

4.2. QUANTUM WALKS

quantum walk, due to the lack of memory between steps.

Quantum walks on graphs are defined as above, with the shifting operator defined intu-

itively in terms of the edge set of the graph, so that direct transitions are allowed only

between vertices connected by an edge. Note that each vertex, or node, can be thought

of as having a group of sub-nodes (or coin positions), each associated with an outgoing

edge. This means that each edge is now associated with two coin positions, one from each

vertex connected by the edge. In effect, this describes a directed graph, as each end of an

edge is defined, and can be manipulated, separately.

For a more formal mathematical definition of quantum walks along graphs, consider an

undirected graph G(V,E), characterised by a set of vertices (or nodes) V = v1, v2, . . .,together with a set of edges E = vi, vj, vk, vl, . . ., being unordered pairs connecting

the vertices. A step of the walk is composed of both a coin operation and a shifting oper-

ation (in either order). Associated with each node vi ∈ V with valency di is a group of di

sub-nodes. Each sub-node is in turn associated with a directed edge outgoing from vi, as

described above. The shifting operator then acts on the extended position space, spanned

by these sets of sub-nodes. Its action consists of swapping the probability amplitudes of

the pair of sub-nodes associated with each edge. For example, representing the amplitude

corresponding to the di-edge connecting vi to vj as |vi, vj〉, and the amplitude correspond-

ing to the di-edge in the opposite direction, connecting vj to vi, as |vj , vi〉, the action of

the shifting operator is defined by

S |vi, vj〉 = |vj , vi〉 . (4.1)

The coin operator acts on the same extended position space, and its action consists of

independently mixing the probability amplitudes associated with each group of sub-nodes

of a given node. For example, given an undirected graph G(V,E) with n vertices and k

edges (i.e. |V | = n and |E| = k), there are 2k sub-nodes, or coin positions, hence the

position space of the walk can be represented by a 2k column matrix. Then the shifting

operator S can be represented as a (2k × 2k) permutation matrix, and the coin operator

C as a unitary (2k × 2k) block diagonal matrix, with each block, or coin, representing a

group of sub-nodes associated with a node. Each step of the walk is then represented by

a unitary (2k × 2k) matrix U , where U = S · C, acting on the state space.

Note that quantum walks involve unitary evolution and are hence reversible. Unlike clas-

sical random walks, they do not approach a stationary probability distribution. Indeed,

58


for graphs with a high degree of regularity or large automorphism group, the resulting

probability distributions often exhibit periodic behaviour.

Observations of probability distributions resulting from quantum walks along various

graphs showed a strong relationship between the complexity of the probability ampli-

tude distribution, and the size of the automorphism group. Moreover, since the amplitude

distributions obtained showed a high sensitivity to minor changes in the graph, it seemed

likely that properties of the probability distribution might serve as an effective tool for

producing a graph certificate that would enable the topological uniqueness of a graph to

be established.

These observations provided the motivation behind looking into a possible GI algorithm

based on quantum walks. Initially, this was with a view to quantum systems (and hence

a quantum algorithm) only. However, while constructing the algorithm it became evident

that for the purposes required here, quantum walks can be implemented efficiently on a

classical computer. Here the term ‘efficient’ is used to indicate that the resources require-

ments for a classical implementation of a quantum walk along a graph scale polynomially

with the size of the graph.

Consider the situation above, with a quantum walk along a simple, undirected graph with

n nodes and k edges. There are exactly 2k coin positions, where k ≤ n2. O(n3) compu-

tational time is sufficient for matrix multiplication between two matrices of size (n × n).

Each step of the walk involves applying the shifting and coin operations to the state vec-

tor, of length 2k. Then given shifting and coin operators both described by (2k × 2k)

matrices, each step of the walk can be implemented on a classical computer in a time that

scales with O(k3), being at most O(n6). In fact, the shifting operator can be implemented

much faster than O(n6). A more straightforward implementation than matrix multiplica-

tion, involving directly rearranging the elements of the state space, has an upper bound

scaling of O(n4). In this case, the permutation of the state space corresponding to the

shifting operator is implemented by sequentially mapping between individual states, via

a temporary section of computer memory. Similarly, the coin operator, a block diagonal

matrix containing n blocks each of size at most n, can also be trivially implemented with

an upper bound scaling of O(n4). Hence quantum walks on graphs, in the form described

above, can be implemented on a classical computer using O(n4) resources. As such, the

methods described in Sections 4.3 and 4.4 will be assumed to be implemented classically.

A corresponding quantum implementation is described in Section 4.5.

59

4.3. PROPOSED GI ALGORITHM

For the purposes of a GI algorithm, the coin operator must not be biased with respect to

the labelling of the vertices. In particular, this means that each block (or ‘coin’) of the

coin matrix must be symmetric, with the form

Ci,j =

a i = j

b i 6= jwhere a, b 6= 0. (4.2)

Since each coin must also be unitary, the Grover coin, having the definition Ci,j = 2d−δi,j is

the only purely real, non-trivial coin satisfying the constraint that symmetry with respect

to labelling is preserved.

4.3 Proposed GI Algorithm

A simple GI algorithm directly employing quantum walks could consist of starting in an

equal superposition of all states, evolving a quantum walk along two graphs for some

fixed number of steps, then comparing the two sets of probability amplitude distributions

obtained. If the probability amplitude at one node of one graph had no corresponding

match for any node of the other graph, the graphs would be necessarily distinct. Consider

two graphs in which each node on one graph has the same number of cycles of every

length as at least one node on the other graph. We term such graphs ‘walk-equivalent’. It

is clear that the method above distinguishes graphs that are not walk-equivalent. Indeed,

such a naive method is quite effective at distinguishing most graphs. However, this simple

method fails to distinguish very similar graphs, including some vertex transitive graphs,

and all strongly regular graphs with the same parameters.

Similar methods have been independently attempted in previous works. Shiau et al. [60]

performed single-particle quantum walks on closed graphs, but they concluded that such

walks fail to identify non-isomorphic strongly regular graphs. Emms et al. [61] introduced

a new matrix representation inspired by quantum walks. This representation appears to

yield different spectra for sets of non-isomorphic strongly regular graphs, but it fails to

distinguish other regular graphs, and general non-regular graphs.

Note that methods which attempt to solve GI in general tend to break down for certain

difficult classes of graphs, most notably strongly regular graphs, pairs of which are com-

monly used as test cases for proposed GI algorithms. A strongly regular graph (SRG) with

parameters (n, d, λ, µ) is a d-regular graph of order n, such that all pairs of adjacent nodes

60


have exactly λ common neighbours, and all pairs of non-adjacent nodes have exactly µ

common neighbours.

So applied in their simplest manner, quantum walks cannot distinguish certain classes

of graphs, including SRG’s with the same parameters. However we have significantly

more freedom available in the construction of such an algorithm. In this case, we can

add certain inhomogeneities into the graphs, with the aim of breaking the symmetry

with respect to the walks that exists between them. Such inhomogeneities could take

a variety of forms, such as self loops or gadgets added to a node, extra nodes added

along an edge, or phase additions to nodes or edges. Of these forms, phase represents a

particularly elegant addition to such walks, in that the connectivity of the graph is left

unaltered, and the definition of quantum walks allows for phase additions to nodes or

edges (directed or undirected) without significant extra computational costs. Phase can

be added equivalently to the coin operator (added to blocks of the coin matrix) or the

shifting operator (added to directed edges), or as a separate third component of a step of

the walk. For example, a φ-phase can be added to a di-edge |vi, vj〉 by altering the action

of the shifting operator from:

S |vi, vj〉 = |vj , vi〉 , to S |vi, vj〉 = eiφ |vj , vi〉 . (4.3)

Although pairs of ‘walk-equivalent’ graphs are often not distinguished using simple quan-

tum walks, if phase additions are made to two or more nodes, cycles originating from,

or involving, these nodes will also interact. The connectivity, or geometric relationship

between such paths then becomes important, and affects the evolution of the walks.

Any GI algorithm as a whole must not be biased with respect to the labelling of the

vertices, since we require identical results to be produced from all possible permutations

of a graph. So if phase is added about certain nodes (termed ‘reference nodes’), we need to

cycle over all possible such selections within the algorithm. Given some arbitrary vertex

of a graph as a reference node, the labelling of this node is considered fixed, however to

preserve the arbitrary labelling of all other nodes, phase additions can only be made to

groups of nodes defined solely in terms of this reference node. The only such groups are

the groups of vertices at each fixed distance from the chosen vertex, and the sets of edges

mapping both within and between each of these groups, shown in Fig. 4.3. Explicitly, for

61


a chosen vertex vi of a graph G(V,E), these are the sets of vertices:

V ai ∈ V, V a

i = v ∈ V : dist(vi, v) = a , (4.4)

for any constant a, where dist(v, vi) is the distance between v and vi, and the sets of edges:

Ebi ∈ E, Ebi = (vj , vk) ∈ E : min [dist(vi, vj),dist(vi, vk)] = b , (4.5)

for any constant b. Distinct phase additions can be made to nodes or directed edges within

each of these divisions, consistently with respect to labelling. If two or more reference

nodes are selected, further divisions can be made, consisting of nodes or edges connecting

these nodes, or at some fixed distance from two or more of them.

Figure 4.3: A given graph of diameter 2, showing all possible groups of phase additions,unbiased with respect to labelling, after fixing the labelling of a single node.

Given two isomorphic graphs, for each possible set of chosen reference nodes on one graph,

there must be at least one corresponding set of reference nodes on the other graph yielding

an identical probability amplitude distribution. The aim of a GI algorithm is then to

implement a scheme of phase additions in which, for all non-isomorphic pairs of graphs,

there are efficiently measurable differences between the resulting probability distributions.

4.3.1 Algorithm

Based on this phase addition scheme, we propose the following GI algorithm.

62


We are given a pair of graphs A and B. For each graph G(V,E) with order n, we initialise

a quantum walk, starting in an equal superposition of all states. We then (arbitrarily)

choose two nodes v1 and v2 on the graph, termed ‘reference nodes’. For each of these two

nodes, consider the groups:

G0 = V 0i , i = 1 or 2

G1 = V 1i , and (4.6)

G2 = V/(G0 ∪G1),

defined only in terms of the reference nodes. These groups are the reference node itself

(G0) and the groups of adjacent nodes (G1) and non-adjacent nodes (G2). Phase additions

are made at each step of the walk to G0, the edges connecting G1 to G2, and to the

edges connecting G1 to G0. The walk is evolved for a number of steps with an upper

limit of 2n, to ensure that the walk samples all areas of the graph, with any differences

in amplitude propagating back to the reference nodes. At each step of the walk, the

probability amplitude associated with the node v1 is recorded. The process is then repeated

for all n2 possible choices of v1 and v2, producing n2 sets of probability amplitudes.

In order to establish a ‘baseline’ measure of the symmetry of each graph individually with

respect to these walks, the sets are compared pairwise. In other words, each set of n

probability amplitudes, corresponding to some pair of reference nodes, is compared to

every other set, building up a comparison table with n4 elements. If the set resulting from

one pair of reference nodes vi and vj matches that resulting from another pair vk and vl,

we write a “0” on the comparison table, in the position (i, j, k, l). Otherwise, if the sets

do not match, this is recorded as a “1”.

For the two given graphs, we then obtain two tables, labelled (A-A′) and (B-B′), repre-

senting the comparison of each graph to its own permutation. A third table (A-B), also

containing n4 elements, is constructed by pairwise comparison of the sets of probability

amplitudes from one graph to those from the other graph. These tables provide a mea-

sure of the relative amplitude distributions resulting from walks along the two graphs. In

effect, the (A-A′) table represents a comparison of graph A to itself (or equivalently to

a permutation of itself), whereas the (A-B) table represents a direct comparison between

the amplitude distributions resulting from graphs A and B.

Comparing the amplitudes of the reference nodes (i.e. partial states) is for our pur-

poses equivalent to comparison of the entire distributions, since differences in parts of

63


the distributions propagate through to all nodes with repeated application of the walks.

In effect, repeated steps of the walk results in a recursive partitioning of the vertex set

based on the relative connectivity of each node from the pair of reference nodes. In other

words, differences between the overall states after some number of steps (manifested as

differences between the amplitudes of some small subset of nodes in each graph) are propa-

gated throughout the entire distribution after repeated steps of the walk, and in particular

propagated to the reference nodes, which are then measured.

If the comparison table (A-B) is different to (A-A′) or (B-B′), the two graphs A and B are

necessarily non-isomorphic. For the purposes of the algorithm, the total number of “1”s in

the three comparison tables (A-A′), (B-B′) and (A-B) are compared. If these totals differ,

the graphs A and B are again necessarily different. The important question is whether

the converse is true - whether equal sets of probability amplitude distributions resulting

in equal comparison tables imply that the graphs are isomorphic. Computational testing

against databases of graphs, detailed in Section 4.4, supports this possibility, and attempts

to establish a formal proof are currently in progress.

4.3.2 Discussion of Algorithm

The use of two reference nodes is a vital aspect of this algorithm. Addition of phase about

only one reference node fails to identify sufficient connectivity information. For instance,

the properties of strongly regular graphs (namely that they are regular, all pairs of adjacent

nodes have the same number of common neighbours, and all pairs of non-adjacent nodes

have the same number of common neighbours) mean that they cannot be distinguished

through the use of a single reference node, since the propagation of phase from this node

will be identical for SRG’s with the same parameters. Specifically, the number of paths

of any given length between two points at some fixed distance is identical, and hence the

propagation of phase throughout the graph, emanating from a single reference node is also

identical.

However given two or more reference points, the complex amplitudes associated with paths

originating from both points will interact, leading to differences in the resulting amplitude

distributions of the walks. Taking further steps of the walks recursively partitions the

vertex set, based on the relative connectivity of the nodes. Essentially, differences oc-

curring within the probability amplitude distributions arising from quantum walks along

two graphs will propagation through the distribution. For instance, if the amplitude at

64


node x is different after some number of steps of the walk, all nodes connected to x will

have different amplitudes after taking further steps of the walk. Expressing the veracity

of the above GI algorithm in an alternate form, it requires that all isomorphism classes of

graphs are distinguished based on the connectivity of nodes relative to all possible pairs

of reference points.

Given differences in the amplitude distributions resulting from walks along a pair of graphs,

these differences will appear in the comparison tables produced by the algorithm, as “1”s

in positions where the “0”s (matches) would otherwise occur when comparing one of the

graphs to a permutation of itself.

For example, consider the two strongly regular graphs with parameters (16,6,2,2), shown

in Fig. 4.4. Running through the above steps, we find that for the first graph, adding

phase about nodes 1 and 2, the amplitude distribution measured at node 1 matches that

obtained for 144 other pairs of nodes of this graph, shown in the comparison table of

Fig. 4.5(a). However, it does not match the amplitudes obtained from any of the other

pairs from the second graph, as shown in the comparison table of Fig. 4.5(b). If these

two graphs were isomorphic this total number of matches would be necessarily identical.

Hence, the graphs are distinguished.

Figure 4.4: The two strongly regular graphs with parameters (16,6,2,2).

4.3.3 Complexity of Algorithm

The complexity of this algorithm can be ascertained by first noting (from Section 4.2) that

the computational time required to implement a step of a quantum walk along a graph

of order n scales with an upper bound of O(n4). Constructing the sets of probability

amplitudes requires taking n steps of the walk for each of the n2 pairs of phase nodes, hence

requiring a total computational time of O(n7). An upper bound of O(n4) comparisons

65


Figure 4.5: Samples of the comparison tables resulting from the two strongly regulargraphs of Fig. 4.4. (a) and (b) represent the (A-A′) and (A-B) comparison tables respectively,from positions (1,2,*,*). If the graphs were isomorphic, these tables would necessarily containthe same number of “1”s and “0”s.

are then made between elements of these sets of amplitudes. Hence the algorithm in its

present form scales with O(n7 + n4) = O(n7).

This complexity calculation, however, does not take into account the possible precision

required for the probability amplitude. It may be possible that after the walk is evolved

for the required number of steps, the differences in probability amplitudes between non-

isomorphic graphs are exponentially small relative to the graph size. Since the information

that is required when comparing a pair of graphs is the relative values of the probability

amplitudes, and not their absolute values, it is possible that this problem can be circum-

vented (at least in the classical case), given appropriate alterations to the algorithm. This

is the subject of further study, and will be left as an open problem at this stage.

4.3.4 Addendum to the Algorithm - Finding an Isomorphism

Since the problems of testing for isomorphic graphs, and finding a specific mapping (iso-

morphism) between isomorphic graphs are Turing equivalent [102], we can extend the

algorithm described above to provide an efficient (polynomial time) method to either find

an isomorphism between graphs which are not distinguished and hence prove they are

isomorphic, or prove that the algorithm does not solve GI.

Specifically, consider two graphs A and B which are not distinguished by the above algo-

rithm. We choose a node of A, and fix its labelling, for instance by adding a self loop or

gadget to the node. We repeat this for one node of B, fixing its labelling in the same way,

66


and then compare the graphs using the above algorithm. We then repeat this method, cy-

cling over all n2 combinations of single nodes in A and B. If the two graphs are isomorphic,

isomorphisms between them will be manifested as matches between the graphs.

Any mapping compiled from such matches can then be applied to the adjacency matrices,

and easily checked. If it does not represent an isomorphism between the two graphs,

then the algorithm cannot distinguish the graphs down to their isomorphism classes. The

benefit of this addendum to the algorithm is that the end result of comparing a pair of

graphs (which is arrived at in polynomial time) must be one of the following: definite

knowledge that the graphs are not isomorphic, definite knowledge that the graphs are

isomorphic, or a proof that the algorithm does not solve GI.

4.4 Testing Algorithm

We tested the above algorithm against a variety of graphs, listed in Tables 4.1 and 4.2,

taken from databases at [104–106]. A wide variety of classes of graphs were chosen, in

order to demonstrate that the algorithm is not limited to a specific type of graph. In

particular, a large number of strongly regular graphs (SRG’s) were examined, as groups

of SRG’s with the same parameters are particularly similar, and often used as test cases

for proposed GI algorithms. In addition, groups of projective planes were chosen as they

again represent a particularly difficult class of graphs to distinguish. In each case, all

graphs were compared pairwise.

In practice, most graphs do not need to be compared directly as described in the previous

section. Instead, given two graphs, each graph can first be compared to its own permu-

tation, with the total number of “0”s recorded as above summed, resulting in a single

integer. This represents the total number of matches obtained from pairwise comparisons

of the probability amplitudes resulting from each pair of reference nodes. If this total

differs between the two graphs, they are again necessarily distinct. Otherwise, they can

then be compared directly. This presents the possibility of developing a succinct graph

certificate using a similar method. Currently, this scheme only yields an incomplete cer-

tificate, in that the final integer obtained is not unique to the graph. However, it is still

quite useful for pairwise comparisons of large groups of graphs. All graphs of each set in

Tables 4.1 and 4.2 were first compared indirectly, via this incomplete graph certificate,

greatly decreasing the required number of direct comparisons. For example, there are

67

4.4. TESTING ALGORITHM

32548 strongly regular graphs with parameters (36, 15, 6, 6), requiring 5.3× 108 pairwise

comparisons. Of these however, only 2 × 105 pairs possessed the same certificate, and

were then distinguished with direct comparisons. All non-isomorphic graphs tested were

successfully distinguished.

Table 4.1: Groups of SRG’s tested (all graphs in each group were compared pairwise).

Parameters Group size

(16, 6, 2, 2) 2(25, 12, 5, 6) 15(26, 10, 3, 4) 10(28, 12, 6, 4) 4(29, 14, 6, 7) 41(35, 18, 9, 9) 227(35, 16, 6, 8) 3854(36, 14, 4, 6) 180(36, 15, 6, 6) 32548(37, 18, 8, 9) 6760(40, 12, 2, 4) 28(45, 12, 3, 3) 168(64, 18, 2, 6) 78

Table 4.2: Groups of other graphs tested.

Graph type Order of graph Group size

Eulerian 8 1849 178210 33120

Cubic vertex-transitive 48 3260 2664 38

Planar 7 646Tree 14 3159Vertex-critical 10 (χ = 4) 2453Edge-critical 12 (χ = 7) 395Vertex-transitive 29 1182Hypohamiltonian 26 2033Projective planes of order 16 273 13Projective planes of order 49 4902 200

It is apparent that there is considerable freedom associated with the chosen phase additions

in such a method. An alternate, simpler phase scheme could involve the addition of

phase, at each step, to only the two chosen reference nodes. Even simpler would be

68


to restrict this phase to a π-phase addition. In effect, restriction to π-phase additions

means that we are no longer working with phases encompassing the one dimensional

complex unit circle, but rather two points - the set ±1. This simplified scheme was

tested against all the graphs of tables 1 and 2, distinguishing all pairs expect one, a

pair of SRG’s with parameters (40, 12, 2, 4). Specifically, these are the point graph of

the generalised quadrangle GQ(3, 3) and its dual (i.e. the point graph and line graph

of GQ(3, 3)). These graphs are particularly similar, being both distance transitive (and

hence rank 3), and having the same size automorphism group. Given that they were an

exception amongst all graphs tested, being the only non-isomorphic pair not distinguished

with the simplest two-phase scheme, several other rank 3 graphs, together with additional

generalised quadrangles were tested. In particular, GQ(5, 5) and GQ(7, 7) (having orders

156 and 400 respectively), together with their respective duals, were compared. Again,

the simplest two-phase scheme was not sufficient to distinguish them, with the original

scheme, involving further phase additions, to edges, required. All other pairs of rank

3 graphs tested, ranging in order from 49 to 364, were distinguished using the simplest

two-phase scheme.

4.5 Quantum Implementation

Up to this point only a classical implementation of a GI algorithm employing quantum

walks has been discussed. However as the name suggests, quantum walks can be very

efficiently implemented on a quantum computer, or some quantum system specifically

designed to implement them. The advantage of a quantum system is the availability of a

state space, or memory that grows exponentially with the number of qubits, allowing in

particular for a quantum walk to be evolved more efficiently than on a classical computer.

There are also disadvantages in the quantum implementation however. The complete

knowledge of the evolution of the probability amplitude distribution is not available for a

quantum system, and instead repeated measurements must be made to approximate the

parts of the wavefunction of interest. Hence any algorithm developed would no longer be

deterministic. In some ways the quantum implementation has less freedom in the possible

properties of the walk, in that it is restricted to be a unitary process. Depending on

the properties of the implementation used however, potentially useful expansions to the

algorithm may be accompanied by significantly less, if any, additional costs, compared to

those associated with a classical implementation.

69

4.5. QUANTUM IMPLEMENTATION

Modifications must be made to this algorithm to implement it within the constraints of a

quantum system. Specifically, since the probability distribution is not directly known, the

measurement process requires some important changes. We can no longer simply record

the probability amplitude at one of the phase nodes at each step. Rather than recording

these amplitudes from each graph separately, then comparing them, the amplitudes from

the two graphs are directly compared. After evolving the walk along each graph separately

for some given number of steps, an additional node is introduced into the system, con-

necting the two nodes on each graph that are to be compared, as shown in Fig. 4.6. The

probability amplitudes in each of these two nodes are shifted into the connecting node,

with a π-phase change applied to the wavepacket entering from one direction. If the con-

nected vertices from each graph have identical probability amplitudes, the π-phase change

will cause them to cancel out exactly, otherwise there will be some non-zero probability

to find the walk at this connecting node.

Figure 4.6: An illustration of the proposed quantum measurement scheme. The amplitudeat two nodes is compared by connecting them via an additional node introduced into thesystem, at which the measurement is made.

The advantage of this method lies in the requirement of only one measurement, which is a

direct comparison between the graphs. If the connected nodes are equivalent with respect

to the walk, there will always be zero probability to measure the walk at the connecting

node, hence the measurement itself will not alter the system. If they are not equivalent,

the effects from the measurement will not matter, as all that is required is a zero or non-

zero measurement. Note that such a measurement will still need to be repeated many

times (or until a non-zero measurement is made) to achieve the desired level of accuracy.

In particular, assume the connected nodes are not equivalent with respect to the walk,

and the average probability of a non-zero measurement is 1p . Assuming this probability

remains constant at each step †, the probability P (m) of a non-zero measurement after m

†Fluctuations will occur, however they will increase rather than decrease the level of accuracy after afixed time, providing this time is larger than the time scale of the fluctuations.

70


steps, is:

P (m) =

m−1∑

i=0

[1

p

(p− 1

p

)i]

≈ 1− e−m/p for large p. (4.7)

Assuming the walk on a graph of order n is on average evenly distributed along all n

nodes, the average probability is simply 1n . In practice, the probability amplitude of the

walks is often concentrated in small subsets of a graph. Accounting for this by taking the

average probability at a given node to be 1nc , for some constant c, we then obtain:

P (m) ≈ 1− em/nc , (4.8)

giving an error after m measurements of approximately em/nc. So to obtain any fixed

accuracy, the number of required measurements scales with O(log(nc)) = O(log(n)).

Considering the scaling of this quantum algorithm, we note that the two steps of obtaining

the sets of amplitudes, and comparing these sets, can no longer be performed separately

within the algorithm. Instead, for each pair of phase nodes on one graph, we must cycle

over all n2 possible pairs on the other graph, for each one comparing the amplitudes as

described above, leading to n4 possible combinations. O(n) steps are required for each set

of parameters. Assuming an efficient implementation of the quantum walks, the time to

execute one step of the walk scales with O(log(n)). The complete algorithm would then

scale with O(n5log(n)2) < O(n6).

As in Section 4.3, this complexity consideration is only correct if the differences between

the amplitudes of non-isomorphic graphs are assumed to scale at worst polynomially with

graph size. Otherwise the 1nc factor is no longer valid, and the complexity of the algorithm

may no longer be polynomial in n.

Note that the actual space explored in the algorithm (via the walk along the graph) in-

volves a memory with an upper bound of n2, where n is the number of vertices. Specifically

the memory required scales with 2e ≤ n(n − 1), where e is the number of edges in the

graph. Since there will only be exactly 2e states accessed by the walk, requiring in theory

only log(2e) qubits in a quantum implementation. This allows a more efficient implemen-

tation of quantum walks than that attainable with the classical method. Nonetheless, the

polynomial growth (with graph size) of the state space of a quantum walk along a graph

is the reason why quantum walks can also be simulated in polynomial time on a classical

71

4.6. CONCLUSION

computer.

4.6 Conclusion

We hope that this work will aid in a deeper understanding of the properties of quantum

walks and how they differ from simple classical random walks. For the purposes of evolving

such walks within a polynomially growing state space, without the presence of noise or

quantum measurements, we see that they are classically efficiently implementable. It is an

interesting question as to which algorithms exist employing quantum walks that cannot

be implemented classically.

With regards to the classical GI algorithm developed in this work, several open questions

remain. The algorithm successively distinguished all non-isomorphic graphs tested, in-

cluding groups of graphs traditionally considered to be ‘difficult’ to distinguish. How to

prove whether it can distinguish general graphs is the focus of further work. The general

methods employed here may also prove beneficial in the construction of other algorithms.

The measurement method utilised by the quantum GI algorithm is quite novel, and rep-

resents an effective means of extracting information from system wavefunctions without

undesirable disturbances to the system.

Possible derivatives of quantum walks, for instance with multiple-step memories, may also

provide fruitful areas of future study. Finally, the key property of quantum walks used

here to distinguish graphs, namely the interactions between all possible paths through the

state space, conceivably has a variety of other uses. For problems in which rapid sampling

of the state space, and control of the interactions between paths through this space are

desired, quantum walks provide promising candidates for their solution.

4.7 Subsequent Work

In the intervening time since the completion of the work described in this chapter, sig-

nificant new results have come to light regarding quantum walk approaches to the graph

isomorphism algorithm. New related approaches have been suggested, for instance the

work of Gamble et al. [17], proposing a quantum walk GI algorithm employing two inter-

acting walkers, with the walkers specifically exhibiting either bosonic or fermionic interac-

tion characteristics. They also test their algorithm against the same database of strongly

72


regular graphs used here, finding similarly that all such graphs are distinguished (at least

in the case of interacting fermions). The similarities between these results are particularly

apparent when the methods are viewed in light of the work of Part II, and will be briefly

discussed there.

Further results are presented by Smith in [16], relating specifically to the work of Gamble

et al. [17]. Guo [62] also considers quantum walk isomorphism algorithms, in this case

specifically along strongly regular graphs, posing the question as to whether quantum

walk based methods suffice to distinguish strongly regular graph alone.

Independently to these works, we have continued our analysis of the algorithm discussed

here. In particular, a conversion to a purely classical algorithm decoupled from quantum

walks was developed (namely a variant of the Weisfeiler-Leman method, to be discussed

throughout Part II) soon after the publication of these results in [19]. This classical

algorithm is conceptually much simpler and more intuitive, and has been the basis for a

large body of existing research spanning almost 40 years.

As a consequence of the discussion in Part II, the quantum walk algorithm developed here

no longer represents a direct attempt at solving the GI problem, in that counterexample

families of graphs are introduced which are provably not distinguished by this method.

However the algorithm and general method of this chapter still has further points of merit.

Firstly, as discussed in Guo [62] and Smith [16], it remains a possibility (and there are par-

ticularly attractive reasons for considering it) that this quantum walk GI algorithm may

suffice for strongly regular graphs. In addition the novel nature of the quantum measure-

ment scheme developed here could potentially be applied to other quantum algorithms. It

essentially acts as an alternative amplitude amplification algorithm; given non-isomorphic

graphs which are distinguished by this method, the amplitude at the connecting node can

be brought arbitrarily close to unity by applying O(log(n)) steps of the walk. As such it

provides an alternative method of comparing the complex amplitude distributions of each

graph - involving a direct, single measurement of a single qubit rather than a series of

measurements to build up information regarding the amplitude distribution. Further, the

frequency analysis of the walk probability distributions identifies potentially useful prop-

erties of quantum walks which can possibly be exploited in related problems (for instance

as in the work of Berry and Wang [15]). The possibility remains that a direct physical im-

plementation of quantum walks can effectively implement the standard Weisfeiler-Leman

73

4.7. SUBSEQUENT WORK

algorithm introduced in Part II more efficiently than classical methods, allowing a polyno-

mial speed-up. Finally, note that the known counterexample graphs described in Part II

share fairly restrictive properties. It is possible that such properties may turn out to be a

requirement for a given graph to be such a counterexample. If so, such graphs may prove

to be in some sense trivial cases, allowing a simple extension to the method described here

to suffice for general graphs.

74

Part II

The Weisfeiler-Leman Method and

Graph Isomorphism Testing

75

Introduction

Based on B. L. Douglas’ paper, The Weisfeiler-Leman Method

and Graph Isomorphism Testing, available at arxiv:1101.5211

[math.CO] and currently under submission.

In this Part the graph isomorphism algorithm of Chapter 4 is generalised to a classical,

well-known graph isomorphism algorithm known as the Weisfeiler-Leman method. Origi-

nally this work progressed independently to the wealth of literature on this subject, and

indeed without knowledge of its existence. Instead, a generalised classical form of the

quantum walk method was developed, as a natural way to implement what was essentially

occurring in the quantum walk method more efficiently. Despite the interference proper-

ties and other novel characteristics of the quantum walk method, in its essence it involves

recursively refining the set of di-edges (i, j) of a graph (associated with the related Ai,j

entry in the adjacency matrix) dependent on the previous colouring of di-edges at dis-

tances 1 and 2, at each step. By backtracking to the start of the process, we see that the

initial colouring simply stems from the marking of the phase nodes, and the degree of each

vertex. Hence rather than relying on the quantum walk to achieve this, it can be directly

implemented without the need for ‘shifting’ and ‘coin’ operators, by instead recursively

updating a set of edge colourings.

In fact the quantum walk method of Chapter 4 will be shown in Section 5.3.1 to be no more

powerful, in terms of characterising graphs, than a closely related variant of the Weisfeiler-

Leman method: namely the depth-2 1-dim WL method described in that section.

Apart from the brief discussion of the relationship between the depth-2 1-dim WL method

and the quantum walk algorithm of Chapter 4, the remainder of this Part will focus purely

on classical algorithms and methods of computation. As the three Parts of this thesis are

76

designed to be potentially readable in isolation, several definitions covered in Part I will be

restated here. Note also that a few of the notational conventions are altered in this section;

all such differences are enumerated in Section 5.1 immediately following the introduction.

The chief consideration of this Part is the application of the Weisfeiler-Leman (WL)

method to the graph isomorphism (GI) problem. Following its introduction in the 1970’s,

this method was considered to be a possible candidate for a solution to the GI problem.

However subsequent, seminal papers by Cai, Furer and Immerman [3], and Evdokimov

and Ponomarenko [4] seemed to eliminate this consideration, presenting families of non-

isomorphic pairs of graphs that the WL method cannot distinguish in polynomial time,

relative to graph size. Indeed, following the work of Cai, Furer and Immerman [3], the

question of whether the WL method or some minor variation might solve GI has (to the

knowledge of the author) been considered closed. However, by analysing the effects of a

slight variant of the WL method constructed here, this work intends to re-open the ques-

tion as to whether the general WL approach might still be used to solve the GI problem.

In particular, in this work we focus on partitioning the vertex set of a given graph to its

orbits, rather than on directly providing a certificate characterising the graph’s isomor-

phism class. We show that although the certificates provided by the WL method do not

efficiently distinguish the counterexample pairs of [3] and [4], under certain assumptions

these graphs are individually refined down to their orbits.

Although this may seem at first to be a trivial distinction, since the inability to produce a

certificate for individual graphs implies the inability to partition certain combinations of

these graphs down to their orbits, we analyse the conditions under which graphs derived

from these counterexample families have been shown to not be efficiently partitioned down

to their orbits by the WL method. We show that in the cases where this trivially occurs,

the graphs of interest possess certain restrictive properties, allowing an extension to the

WL method to partition these graphs down to their orbits, and thus allowing a recursive

WL method to distinguish them.

The main result of this work is the proposal of an extension to the recursive k-dim WL

method, for the purposes of dealing with known counterexamples. By first enumerating

such known counterexamples, it is established that these graphs possess fairly restric-

tive properties, particularly in relation to their requisite sub-constituents. Given such

a graph, the proposed extension involves first applying a decomposition method which

isolates relevant subgraphs. This decomposition method is recursively applied, yielding

77

sub-constituents which can be then characterised by the recursive k-dim WL method. The

method then backtracks through these decomposition steps, and at each step it is proven

(under certain assumptions regarding the initial graph) that the corresponding graph is

characterised.

Although it is proven that these assumptions hold for the known counterexample graphs

constructed in [3] and [4], hence dealing with the currently known counterexamples, the

significance of this extension lies in the fact that it is not limited only to these token

counterexamples, but is instead applicable to a wide range of potential counterexample

graphs, to which these two examples belong.

Far from providing convincing arguments that the general Weisfeiler-Leman approach can

be used to solve GI, the aim of this work is merely to argue that this remains a possibility,

despite the ground-breaking results of [3] and [4].

The structure of this Part is as follows: Chapter 5 provides some background to the graph

isomorphism problem in general, and the Weisfeiler-Leman method in particular, and in

Section 5.2 we consider a small subset of previous work done in this area. A formal

description of the WL method is then provided in Section 5.3, discussing and deriving

some basic properties of this method, and in this context highlighting and discussing

some of the major counterexample results of [3, 4] in Sections 5.3.2 and 5.4. Following

this introductory material, Chapter 6 contains the new results provided by this work.

In Section 6.1 the properties of general graph extensions (of which the graph families

of [3] and [4] are examples) are explored, and we show that the ability of the recursive

k-dim WL method to distinguish graphs is invariant under such extensions, given certain

assumptions. In Section 6.3 some relevant properties of the WL method are derived. In

Section 6.4 an extension to the recursive k-dim WL method is presented and is shown to

successfully characterise all known k-equivalent graphs, given certain assumptions.

78

Chapter 5

Background and Known

Counterexamples

5.1 Graph Theoretic Notation

For a graph G, we denote the vertex and edge sets of G by V (G) and E(G) respectively. G

represents the complement of G, in which edges and non-edges are switched. Let v ∈ V (G).

Then d(v) and e(v) denote the sets of neighbours and non-neighbours of v, such that

d(v) = x ∈ V (G) : v, x ∈ E(G), and similarly

e(v) = x ∈ V (G) : v, x /∈ E(G).

The valency of v is the number of edges incident with v, namely |d(v)|. We will generally

be dealing with undirected graphs. Where this is not the case, the neighbour set d(v)

includes di-edges incident with v oriented in either direction. If H is a proper subgraph

of G, denoted H ⊂ G, sets S in (resp. operations on) G whose extent (resp. action) is

restricted to H will be denoted by S |H , or S restricted to H. Given some property or

value c held by some members of a set S, the set of all members of S with the property

c is denoted by [c]. For instance, the set of all vertices (within some implicit set S) with

colour class c is [c]. Alternatively, when c is a positive integer, [c] denotes the set 1, . . . , c.Where S ⊂ V (G) for some graph G, the subgraph of G induced on S will simply be referred

to as S ⊂ G, where the question of whether S denotes a set of vertices or a graph will

be clear from the context where not explicitly stated. Similarly, G\H denotes either the

relevant induced subgraph or the vertex set, depending on the context. Vertex-disjoint

79

5.2. THE GRAPH ISOMORPHISM PROBLEM

subgraphs S1, S2 ⊂ G have the property V (S1) ∩ V (S2) = ∅. The distance between two

vertices u, v ∈ V (G) is defined as the length of the smallest path connecting them, and

denoted dist(x, y).

A separator of a graph G is a subset S ⊂ V (G) such that G\S has no connected com-

ponents of size |V |/2 or larger. The separator size of G is the size of the smallest such

separator. The join of two graphs G1 and G2 is the graph H = G1 ∪ G2 in which

V (H) = V (G1) ∪ V (G2) and E(H) = E(G1) ∪ E(G2) ∪A, where

A = x, y ∈ V (H)× V (H) : x ∈ V (G1), y ∈ V (G2). (5.1)

5.2 The Graph Isomorphism Problem

5.2.1 A General Background

The question of efficiently determining whether two given graphs are isomorphic is a long-

standing open problem in mathematics. It has attracted considerable attention and effort,

due both to its practical importance and its relationship to questions of computational

complexity. Examples of excellent reference articles that provide a more thorough back-

ground to the GI problem can be found in [23], [107] and [108].

The exact complexity status of the graph isomorphism (GI) problem remains unknown. It

is known to be in the class NP, however neither an NP-completeness proof or a polynomial

time solution have been found. It is generally considered unlikely to be in NP-complete

[109,110], in part because the corresponding testing and counting problems are polynomial-

time equivalent, unlike the apparent case for all other known NP-complete problems.

Further supporting evidence is provided by Schoning [111], who demonstrates that GI is

not NP-complete unless the polynomial-time hierarchy collapses. As such it provides a

promising candidate for a problem that is neither in P nor NP-complete.

Efficient GI algorithms do exist for several restricted classes of graphs, such as trees [96],

planar graphs [112] and graphs with certain bounded parameters, including valence [113],

eigenvalue multiplicity [114] and genus [115]. The GI problem has several additional,

relevant properties. It is generally easy to solve in practice, and for many, if not most

practical applications the GI problem can be viewed as solved, in that the types of graphs

involved can be efficiently characterised by existing algorithms, such as Brendan Mackay’s

‘Nauty’ package [99]. It is also easy to solve for almost all graphs [20, 21]. However the

80

CHAPTER 5. BACKGROUND AND KNOWN COUNTEREXAMPLES

best current GI algorithm for general graphs has an upper bound of O(e√nlogn) [22–24].

Hence the interest in GI lies largely in its complexity status, it being one of the interesting

problems where practical and theoretical notions of efficiency do not coincide.

GI is polynomial time equivalent to several related problems, including finding an isomor-

phism map between graphs, if it exists, and determining either the order, generators or

orbits of the automorphism group of a graph [116]. Proposed algorithms to distinguish

graphs generally fall into two main (not necessarily disjoint) categories: combinatorial

and group theoretic. Here we will be discussing a common type of combinatorial method,

based on the iterative vertex-classification (or vertex-refinement) method, and collectively

termed the Weisfeiler-Leman method.

5.2.2 The Weisfeiler-Leman method

The general type of method labelled as iterative vertex classification is discussed in [108]

and [23].

Perhaps the simplest such method begins by partitioning the vertex set of a graph (or

equivalently colouring the vertices) according to vertex valency. Then at each subsequent

step the colour of each vertex is updated to reflect its previous colour together with the

multiset of colours of its neighbours. This proceeds iteratively until a stable colouring (or

equitable partition) is reached. This method is also known as the 1-dimensional Weisfeiler-

Leman method. The history and details of the generalised k-dimensional Weisfeiler-Leman

method (which we will term the k-dim WL as in [3] and [117]) and other related methods

can be found in [118–120], among others. Although conceptually quite simple, the 1-dim

WL method succeeds in characterising almost all graphs in linear time [21], although it

cannot for instance partition the vertex set of regular graphs. The sorted set of vertex

colouring resulting from the above process are an example of a graph invariant, defined

as a property or value related to a graph with form irrespective of the vertex labelling. A

complete graph invariant partitions graphs into their isomorphism classes. Hence a given

invariant must take the same value for isomorphic graphs, however non-isomorphic graphs

are not necessarily distinguished, unless it is a complete invariant.

In the k-dim WL method, we instead start with k-tuples of the vertex set, colouring

them according to their isomorphism type. At each step the set of k-tuples is further

partitioned by considering the ordered multiset of colours of the ‘neighbours’ of a given

k-tuple (here the neighbours are the k-tuples differing in exactly one element). Again, this

81

5.3. FORMAL DESCRIPTION OF WL METHOD

is repeated until an equitable partition is reached. Following its introduction by Weisfeiler

and Leman in [119], the general k-dim WL method and related methods have reappeared

several times, and in various forms. For instance Audenaert et al. [121] proposed a graph

isomorphism method based on the symmetric powers of the adjacency matrix of a graph,

which was later shown in Alzaga et al. [122] and Barghi and Ponomarenko [123] to be no

more effective than the k-dim WL method. Similarly, GI algorithms based on quantum

walks have been proposed in Douglas [19] and Gamble et al. [17]. The algorithm of [17]

was shown in Smith [16] to be no stronger than the k-dim WL, while we show here that

the algorithm of [19] is trivially no stronger than a variant of the k-dim WL method, which

we term the depth-(k − 1) 1-dim WL method, and define in Section 5.3. One significant

aspect of the WL method alluded to by its continuing reappearance in varying forms is

its intuitive, in some ways natural, combinatorial form.

A more formal definition is given in Section 5.3, however at this point it is clear that

just as the 1-dim WL fails for regular graphs, providing no useful information beyond the

graph’s order, the 2-dim WL will fail for strongly regular graphs. Similarly, the k-dim WL

method cannot partition the vertex set of k-isoregular graphs (alternatively k-tuple regular

graphs), defined as in [124] to be graphs in which the number of common neighbours of

any k-tuple of a given isomorphism type is constant (for instance, strongly regular graphs

are 2-isoregular). The results of Gol’fand and Klin [125] and Cameron [124], classifying

5-isoregular graphs to a few trivial cases, and proving that 5-isoregular graphs are k-

isoregular for all k, in part supported the conjecture that the k-dim WL method might,

with k some small constant, suffice to classify all graphs.

The k-dim WL method can be implemented for a graph on n vertices in time O(nk+1),

hence if even the O(log(n))-dim WL method sufficed to distinguish all graphs on n vertices

it would solve GI. However the results of [3, 126] disposed of this possibility, providing

examples of a family of pairs of graphs with O(n) vertices which the (n − 1)-dim WL

method failed to distinguish. This situation was explored further in [4] and [123], in terms

of coherent configurations, with an additional family of counterexample graphs presented.

5.3 Formal Description of WL Method

Let G be an edge- and vertex-coloured graph, where |V (G)| = n, and for any u, v ∈ V (G)

such that (u, v) ∈ E(G), ω(u) and ω(u, v) denote the colouring of the vertex u and edge

82


(u, v) respectively. Consider the set V (G)k of k-tuples of V (G). The k-dim WL method

proceeds iteratively, with the colour of all k-tuples being updated at each step. Given an

ordered k-tuple S = (v1, v2, . . . , vk) ∈ V (G)k, consider the ordered set of ‘neighbouring’

k-tuples

S′(x) = ((v1, . . . , vk−1, x), . . . , (x, v2, . . . , vk)), x ∈ V (G). (5.2)

Then after t steps of the k-dim WL method, the colour of S ∈ V (G)k is denoted by

WLtk(S), such that

WL0k(S) = iso(S), and

WLtk(S) = 〈WLt−1k (S), SortWLt−1

k (S′(x)) : x ∈ V (G) 〉, (5.3)

where ‘iso(S)’ denotes the isomorphism class of the ordered k-tuple S, such that for S1 =

(x1, . . . , xk), S2 = (y1, . . . , yk) ∈ V k, we have iso(S1) = iso(S2) if and only if for all i, j ∈ [k]

the following hold:

1. xi = xj if and only if yi = yj .

2. (xi, xj) ∈ E(G) if and only if (yi, yj) ∈ E(G) and ω(xi, xj) = ω(yi, yj).

3. ω(xi) = ω(yi).

Note that the sorting function ‘Sort’ used throughout this chapter applies only to the

outermost dimension of nested lists, unless otherwise stated. In particular, it does not alter

the internal ordering of each individual ordered set comprising S′(x). As an additional

note regarding notation, the angle brackets enclosing the right hand side of Eq. (5.3) are

used in the style of [3] to delimit an ordered set. They are used here to take the place

of round brackets (which denote ordered sets elsewhere in this work) simply for aesthetic

purposes, and this convention will be continued when describing k-dim WL colour classes

as above †.

At each step in the process the WLtk(S) multisets are sorted lexicographically then assigned

a number from 1 to n denoting the new colour class of S, together with a decoding table

to store the remaining information for the purposes of constructing a certificate for the

graph at the end. The algorithm stops when the colouring of k-tuples is stable; when a

†Specifically, this notational convention will only be used to enclose the definition (or reference to thedefinition) of such colour classes.

83


further iteration of the method does not partition the set of k-tuples further. Let this

occur after r steps, such that for all S1, S2 ∈ V (G)k,

WLr+1k (S1) = WLr+1

k (S2) if and only if WLrk(S1) = WLrk(S2). (5.4)

Then the final colouring of k-tuples is denoted WL∞k (S), or simply WLk(S), such that

WLk(S) = 〈 SortWLrk(S′(x)) : x ∈ V (G) 〉. (5.5)

Hence for graphs G and H, WLk(G) = WLk(H) if and only if there exists a bijection

mapping V (G) to V (H) preserving the colouring of k-tuples.

Note that given some constant k and a graph on n vertices, this stable colouring (also

known as the equitable partition) will be reached in O(poly(n)) time. Hence if the k-dim

WL method succeeded in partitioning all graphs down to their orbits (for some constant

or slowly growing k), it would solve the GI problem.

Given the stable partitioning of V k, a corresponding partitioning of t-tuples, for any t < k

can be constructed, such that two t-tuples are assigned identical colours if and only if they

cannot possibly be distinguished based only on the colouring of k-tuples. The process for

colouring the t-tuples x = (x1, . . . , xt) as WLk(x) is detailed in Section 6.3, in which the

following recursive relation is derived:

WLk(x) = 〈 SortWLk(x, i) : i ∈ V (G) 〉, (5.6)

where (x, i) denotes the ordered (t + 1)-tuple (x1, . . . , xt, i). Similarly, a colouring of t-

tuples for t > k can be constructed, again defined recursively by considering the ordered

set of (t− 1)-tuples contained in the t-tuple of interest.

In terms of notation, where the t-tuple x belongs to both G and some subgraph of interest

H ⊂ G, the colouring WLk(x) may refer to the colouring of the t-tuple within either G

or the induced subgraph H. Which it refers to will either be clear from the context or

specified by the notation WLk(x)|G or WLk(x)|H , meaning the t-tuple colouring is relative

to the k-tuple colourings of G or H respectively.

Several closely related variants on the k-dim WL method have been proposed. One such

variant, appearing in [118], employs what could be described as a depth-first approach to

the WL method. It is introduced under the umbrella term of ‘deep stabilisation’ in [118],

84


and involves stabilising a k-tuple followed by applying the 1-dim WL method, then cycling

over all possible such k-tuples. We will term this method the depth-k 1-dim WL method,

and note briefly that it is in some ways analogous to the (k + 1)-dim WL method, and

might be expected to have similar refinement power. Indeed, all relevant results here

regarding the k-dim WL method can be extended to the depth-(k−1) 1-dim WL method.

As mentioned, for some time the k-dim WL method, with sufficiently small k (e.g. where

k = O(log(n)) or even where k is a constant) was thought to represent a potential candidate

for the solution to the GI problem. Then Cai, Furer and Immerman [3] introduced a

family of counterexample graphs (we will term them the CFI counterexamples) for which

the entire global properties of the graph could not be characterised using a sufficiently low

dimension WL method. Specifically, they constructed pairs of non-isomorphic graphs on

O(n) vertices that were distinguished by the n-dim WL method, but not by the (n−1)-dim

WL method.

5.3.1 The Quantum Walk Phase Method

This section will briefly discuss the quantum walk approach to graph isomorphism intro-

duced in Chapter 4, relating it to the depth-2 1-dim WL method. Firstly we will recall

the particulars of this method. Given some graph G, with vertex set V (G) and edge set

E(G), a quantum walk is initialised in an equal superposition of all states. These states

correspond to di-edges of G, in that each vertex v ∈ V (G) is split into |d(v)| states, each

corresponding to an ordered pair (v, x) where v, x ∈ E(G).

An unbiased walk is set up on this graph, using a Grover coin at each vertex state. The

walk is then perturbed by altering the shifting operator (or equivalently the coin operator)

relative to two vertices, with a phase perturbation applied to every di-edge in the graph

according to its distance from each of these ‘phase vertices’. Two cases are considered,

in which the two phase vertices are either distinguishable or indistinguishable, essentially

corresponding to considering perturbations based on the ordered or unordered pair of

distances from each phase node respectively. At this point aspects of this scheme display

close similarities to the WL method described above.

The initial perturbation of the evolution operator, contingent on the position of each di-

edge relative to the phase vertices, is analogous to an initial colouring of each di-edge in

G in the following sense. The graph is either set-wise stabilised relative to the two phase

vertices, or point-wise stabilised twice relative to each (depending on whether they are

85


considered distinguishable or not), after which the initial perturbation described above

corresponds exactly to applying the 1-dim WL method to this stabilised graph†. We will

label the graph G with individualised phase nodes by G′.

Given this perturbed evolution operator, we then apply O(n) steps of the walk. The

amplitude distribution resulting essentially forms a graph invariant. In making use of

this invariant, although the amplitude distribution itself cannot be directly determined,

the measurement method of Section 4.5, essentially compares the (ordered) amplitude

distributions of two such graphs, G and H, determining (to an arbitrarily high accuracy)

whether any differences exist.

What remains is to iterate this procedure, testing all possible relative labellings of V (G)

and V (H), and all possible pairs of phase nodes on each. Denote the probability amplitude

at a given di-edge (u, v), after some fixed number of steps, of O(n), by A(u, v). Note that

the essential role of the quantum walking here is to (in the best case scenario) ensure that

at the point at which the measurement is made, the probability amplitude at any two

given di-edges (u, v) and (x, y) satisfies the property

A(u, v) = A(x, y) if and only if WL1(u, v) |G′= WL1(x, y) |G′ . (5.7)

In particular, note that

A(u, v) 6= A(x, y) only if WL1(u, v) |G′= WL1(x, y) |G′ (5.8)

must hold, and hence the process of evolving the quantum walk along G′ cannot partition

the set of di-edges further than the 1-dim WL method.

So the choice of phase nodes is equivalent to individualisation of these two vertices, and

relative to this stabilised, or coloured graph, the evolution of the perturbed quantum walk

is at best equivalent to applying the 1-dim WL method. Hence the composite of these two

steps corresponds to the depth-2 1-dim WL method described above, and in the best-case

scenario where Eq. (5.7) holds, the quantum walk algorithm of Chapter 4 distinguishes

precisely those non-isomorphic graphs distinguished by the depth-2 1-dim WL method.

The natural question remains as to whether the extension to the recursive k-dim WL

method described in Section 6.4 can similarly be applied to this quantum walk method. To

†Equivalently, from the coherent configuration viewpoint discussed in Section 5.4, this perturbation cor-responds exactly to calculating the cellular closure of G subsequent to individualising the ‘phase’ vertices.

86


address this question, we firstly note that the general properties of the k-dim WL method

derived in Section 6.3 also apply to the depth-(k − 1) 1-dim WL method (or indeed the

depth-(k − i) i-dim WL method). As such, the theorems regarding the decomposition

method of Section 6.4 can similarly be formulated relative to the colour classes originating

from the recursive depth-(k − 1) 1-dim WL method.

Hence it is expected that a similar recursive variant of the quantum walk method can

be used as a basis for this decomposition method. Whilst this would be an indirect,

impractical way of achieving this decomposition, it may prove to be of some interest to

studies of quantum walks.

5.3.2 CFI Counterexamples

In the work of [3], pairs of non-isomorphic graphs with O(k) vertices which cannot be

characterised using the k-dim WL method were constructed from graphs with separator

size k+ 1. In particular, given a graph G they define a related graph X(G) (to be termed

CFI(G) for the remainder of this work), in which each vertex v ∈ V (G) of valency k is

replaced by the graph CFI(v), defined (as in [3]) by the relations:

V (CFI(x)) =A(v) ∪B(v) ∪M(v), where A(v) = ai : 1 ≤ i ≤ k,

B(v) = bi : 1 ≤ i ≤ k, and

M(v) = mS : S ⊆ 1, . . . , k, |S| is even, (5.9)

E(CFI(x)) =(mS , ai) : i ∈ S ∪ (mS , bi) : i /∈ S.

The middle vertices M(v) of CFI(v) are coloured differently to the other vertices (A(v) ∪B(v)). Hence each vertex v of degree k is replaced by a graph of size 2k−1 + 2k, consisting

of a ‘middle’ section (the M(v) vertices) of size 2k−1, and k ai, bi pairs of vertices

representing the endpoints of each edge incident with v, such that each ai, bi pair,

1 ≤ i ≤ k, is associated with some edge v, u (u ∈ V ) incident with v. Furthermore, for

each edge u, v ∈ E(G), the (ai, bi) pairs associated with the endpoints of u, v, termed

au,v, bu,v and av,u, bv,u respectively, are connected, such that au,v is connected to av,u,

and bu,v is connected to bv,u.

Similarly, a graph CFI′(G) is defined as above, however with a single

au,v, av,u, bu,v, bv,u edge pair ‘twisted’, in that these two edges are replaced by

the edges au,v, bv,u, bu,v, av,u. Basic properties of these graphs are discussed in detail

87


in [3] (also see [127] and [117] for additional details). Here we recall some pertinent

results.

Lemma 5.3.1. |Aut(CFIk)| = 2k−1. Each g ∈ Aut(CFIk) corresponds to interchanging

ai and bi for each i in some subset S of 1, 2, . . . , k of even cardinality.

Lemma 5.3.2. Given a graph G, consider a graph CFI ′′(G) defined as in

CFI(G) and CFI ′(G) above, however with t edges twisted. Then CFI ′′(G) ∼=CFI(G) iff t is even, and CFI ′′(G) ∼= CFI ′(G) iff t is odd.

Corollary 5.3.3. CFI(G) CFI ′(G).

Most importantly, the work of [3] proved that given a graph G with separator size k +

1, the k-dim WL method cannot directly distinguish the graphs CFI(G) and CFI′(G).

Specifically, while performing the k-dim WL method on CFI(G) and CFI′(G), at each

step the lexicographically sorted multisets of colours are identical.

A similar family of graphs were constructed in [4], using different methods. They introduce

the term ‘k-equivalent graphs’ to describe graphs that the k-dim WL method cannot

distinguish. Although this definition trivially encompasses isomorphic graphs, when the

term is used here it will refer solely to cases where at least two non-isomorphic graphs

exist that cannot be distinguished by the k-dim WL method. Explicitly, the following

terminology will be used.

Definition 5.3.4. A graph G is termed k-equivalent if:

• There exists a graph H, G H such that WLk(G) = WLk(H).

• k is the largest integer for which this is true (a k-equivalent graph is not (k − 1)-

equivalent, hence cannot be (k + 1)-equivalent).

It will be convenient to make one exception to the terminology that a graph is k-equivalent

only if k is the largest such integer, namely for the definition of a non-k-equivalent graph.

Definition 5.3.5. A graph G is non-k-equivalent if there does not exist a graph H, G H

such that WLk(G) = WLk(H).

Definition 5.3.6. Two k-equivalent graphs G1 and G2 are mutually k-equivalent if

WLk(G1) = WLk(G2).

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Mutually k-equivalent graphs are not necessarily non-isomorphic, however they will be

assumed to belong to a k-equivalence class containing at least two distinct isomorphism

classes. Namely, if G1 and G2 are mutually k-equivalent, and G1∼= G2, then ∃ H G1

such that H and G1 are mutually k-equivalent.

A related concept to be employed in the following work is that of k-similarity.

Definition 5.3.7. Two graphs G and H are termed k-similar if WLk(G) = WLk(H).

This is denoted by G ∼ H (where the relevant k will be clear from the context).

Hence mutually k-equivalent graphs are n-similar for all n ≤ k†, and isomorphic graphs

are n-similar for all n. Note that these concepts of k-similarity and k-equivalence were

used in [123] (and implicitly in [122]) to demonstrate that any two k-similar graphs have

identical k-th symmetric powers, hence addressing the proposition of [121].

Definition 5.3.8. Two subgraphs S ⊂ V (G), R ⊂ V (H) satisfy the relation S ∼k R if

and only if WLk(S) |G= WLk(R) |H . Note that S ∼k R only if G ∼k H.

In the limit where S ∼k R for all k > 0, k-similarity becomes isomorphism, and is denoted

by S ∼ R.

5.4 Coherent Configurations

As it was first proposed in the work of [119], the Weisfeiler-Leman method takes the form

of a matrix algebra associated with a graph, termed the cellular algebra, and later known

as the adjacency algebra (or basis algebra) of a coherent configuration. This section will

provide a brief background to coherent configurations, introducing a further set of k-

equivalent graphs based on coherent configurations. For a more thorough background to

coherent configurations and their relation to the WL method, see [4, 120,123,128].

5.4.1 Definitions

Let V be a finite set, and R = R1, . . . , Rs be a partition of V ×V , such that each Ri ∈ Ris a binary relation on V . A coherent configuration on V is a pair C = (V,R) satisfying

the following conditions:

†And explicitly are not n-similar for all n > k.

89

5.4. COHERENT CONFIGURATIONS

1. There exists a subset R0 of R partitioning the diagonal ∆(V ) of the Cartesian

product V × V .

2. Ri ∈ R if and only if its transpose RTi is in R.

3. Given Ri, Rj , Rk ∈ R, for any (u, v) ∈ Rk, the number of points x ∈ V such that

(u, x) ∈ Ri and (x, v) ∈ Rj is equal to cki,j , independent of the choice of (u, v) ∈ Rk.

The numbers cki,j are called the intersection numbers of C, and the elements of V and Rare called the points and basis relations of C respectively. Similar to adjacency matrices of

graphs, a basis relation Ri can be represented in matrix form by the basis matrix A(Ri),

where:

A(Ri)x,y =

1 if (x, y) ∈ Ri0 otherwise.

(5.10)

Then the coherent configuration conditions above take the form:

1.t∑

i=1

Si = 1|V |, the identity matrix, where R0 = S1, . . . , St.

2. If Ri ∈ R there exists a relation Rj ∈ R such that A(Ri)T = A(Rj).

3. For each i, j ∈ [s], A(Ri)A(Rj) =s∑

k=1

cki,j A(Rk).

The algebra spanned by the A(Ri) is called the adjacency algebra or basis algebra of the

coherent configuration C.

Consider the set of basis relations R0 = S1, . . . , St such that (x, y) ∈ Si only if x = y.

Note that by condition (1),

(x, x) ∈ Ri if and only if u = v, ∀ (u, v) ∈ Ri. (5.11)

The t sets Fi ⊂ V such that Fi = x ∈ V : (x, x) ∈ Si are called the fibres of C. By

condition (1) they form a partition of V .

5.4.2 Weak and Strong Isomorphisms

Two coherent configurations C = (V,R) and C′ = (V ′,R′) are isomorphic (or strongly

isomorphic) if there is a bijection mapping V to V ′, preserving the basis relations. The

90


coherent configurations C and C′ are similar (or weakly isomorphic) if there exists a bijec-

tion ϕ : R → R′, where ϕ : Ri 7→ Rϕ(i), such that

cki,j = cϕ(k)ϕ(i),ϕ(j), for all i, j, k ∈ [s]. (5.12)

Clearly, all strong isomorphisms induce weak isomorphisms, however the converse does

not hold.

5.4.3 Coherent Configurations of Graphs

The set of coherent configurations on V forms a lattice with respect to inclusion [128]. In

particular, given a set of binary relations S1, . . . , St, where each Si ∈ V × V , denote

[S1, . . . , St] to be the smallest coherent configuration C = (V,R) with the property:

For each Si, ∃ a set R1, . . . , Rx ⊂ R such that Si =x⋃

j=1

Rj . (5.13)

[S] is also termed the cellular closure of the set S of binary relations. We define the

coherent configuration associated with an uncoloured graph G to be [G] := [V,E, (V ×V )\E], the smallest coherent configuration in which the vertices, edges and non-edges are

each partitioned by basis relations. Similarly, for an edge- and vertex-coloured graph G,

consider the initial binary relations of G to be the sets of vertices (and edges) of each

colour, together with the set of non-edges, resulting in a corresponding definition for [G].

In fact, the WL method was originally defined in [119] as a way of calculating the adjacency

matrix of the coherent configuration associated with a graph. Specifically, consider a

coloured graph G, in which c(v) denotes the colour of vertex v ∈ V (G), and c(u, v) the

colour of edge (u, v) ∈ E(G).

Theorem 5.4.1 ( [119]). G is the adjacency matrix of a coherent algebra if and only if

G is stable relative to the 1-dim WL method, in that for all u, v ∈ V , c(u) = c(v) only if

WL1(u) = WL1(v).

Analogous to the conversion of k-tuple colourings to 1-tuple colourings described in Section

6.3, the set of basis relations of a coherent configuration C = (V,R) induce a colouring

of the 1-tuples of V , corresponding exactly to the subset R0 of R that partitions ∆(V ).

Denote this colouring of 1-tuples by C, the closure of C.91


A set R of binary relations on V is termed 1-closed if [R] = (V,R). Similarly a graph

is termed 1-closed if it is stable with respect to the 1-dim WL method - if the adjacency

matrix of the graph correspond to that of a coherent configuration. Strongly regular graphs

are trivially 1-closed, as their sets of vertices, edges and non-edges satisfy all conditions of

a coherent configuration (equivalently, their vertex sets are not refined by the 1-dim WL

method).

5.4.4 m-Extended Coherent Configurations

Given a scheme C = (V,R), let Cm = C ⊗ . . . ⊗ C denote the m-fold tensor product of C,and ∆m denote the diagonal of V m = V × . . .× V . Then the m-extension of C is defined

as:

C(m) = [Cm,∆m]. (5.14)

An isomorphism ϕ : C(m) → (C(m))′ is termed an m-isomorphism mapping C to C′. A

similarity (weak isomorphism) from C(m) to (C(m))′ is termed an m-similarity mapping Cto C′.

C(m) denotes the colouring of 1-tuples of V associated with C(m), termed the m-closure

of C. A coherent configuration C (resp. a set of basis relations R) is termed m-closed if

C = C(m) (resp. if [R] = [R](m)). Similarly a graph is m-closed if it is stable with respect

to the m-dim WL method.

Theorem 5.4.2 ( [4]). Denote the orbit partition of a graph G by P. Then for some n,

[G] ≤ [G](2) ≤ . . . ≤ [G](n) = . . . = P. (5.15)

In [4], pairs of non-isomorphic k-similar coherent configurations are constructed for all k.

These coherent configurations are related to cubic graphs with minimum separator size of

3k + 1 or larger, similar to the case for the counterexample graphs of [3].

5.4.5 Examples of Non-Isomorphic k-similar Coherent Configurations

Here we give a brief description of the k-similar, non-isomorphic coherent configurations

constructed in [4], corresponding closely to the definition given in [123]. Associated with

92


these will be pairs of k-equivalent (edge-coloured) graphs which will be analysed together

with the k-equivalent graphs of [3] in depth in later sections.

Let G be a cubic graph on s points. Define a coherent configuration C = (V,R) on 4s

points in the following way. Denote the vertex set of G by I = 1, . . . , s, and associate

with each i ∈ I a fibre Vi of size 4 in C, such that C has exactly s fibres, each of size 4.

For each Vi = 0, 1, 2, 3, let CVi be the coherent configuration on 4 points with the three

non-reflexive basis relations:

E1 = (0, 1), (1, 0), (2, 3), (3, 2), E2 = (0, 2), (2, 0), (1, 3), (3, 1) and

E3 = (0, 3), (3, 0), (2, 1), (1, 2). (5.16)

As Vi is a fibre of C, Ri,i contains 4 basis relations, where

Ri,j = R ∈ R : R ⊂ Vi × Vj. (5.17)

For i 6= j, let

|Ri,j | =

2 if i, j ∈ E(G)

1 otherwise.(5.18)

In the cases where i, j ∈ E(G), define Ri,j as follows.

Assign to each v ∈ d(i) the numbers c(i, v) ∈ 1, 2, 3 with the property:

u, v ∈ d(i) such that u 6= v only if c(i, u) 6= c(i, v). (5.19)

Ri,j consists of two distinct relations, labelled R1,2, with the di-edge (i, j) relative to which

they are defined left implicit. These relations R1, R2 ∈ Ri,j , i, j ∈ E(G) are defined as:

R1 = 〈c(i, j)〉 × 〈c(j, i)〉 ∪ 〈c(i, j)〉 × 〈c(j, i)〉, (5.20)

R2 = (Vi × Vj)\R1, (5.21)

where 〈c(i, j)〉 = 0, c(i, j) ⊂ Vi and 〈c(i, j)〉 = Vi\〈c(i, j)〉.

For any cubic graph G the coherent configuration C = (V,R) described above is called a

Klein scheme associated with G. Further, for each i ∈ I, consider the mapping ψi : R →93


R, where

ψi(R) =

(Vi × Vj)\R if R ∈ Ri,j , and j ∈ d(i)

(Vj × Vi)\R if R ∈ Rj,i, and j ∈ d(i)

R otherwise.

(5.22)

Theorem 5.4.3 ( [4,123]). ψi is an involutory weak isomorphism from C to C not inducing

a strong isomorphism. Further, if G has minimum separator size l > 3k, ψi(C) are C are

k-similar.

5.4.6 The Associated Graph

Given a Klein scheme C = (V,R) associated with some cubic graph G, we can define an

edge-coloured di-graph K(G) associated in turn with C, in the following manner†.

Let V = V1, . . . , Vi as above, and let V (K(G)) = V and E(K(G)) be denoted by E.

Denote the colour of the di-edge (x, y) ∈ E by C(x, y). Then given x ∈ Vi, y ∈ Vj ,

(x, y) /∈ E if and only if i 6= j and i, j /∈ E(G). (5.23)

Further, denote the colour of the di-edge (x, y) ∈ E by C(x, y). Then

C(u, v) = C(x, y) if and only if (u, v), (x, y) ∈ R, for some R ∈ R. (5.24)

Hence di-edges are assigned the same colour only in the case where they belong to the

same basis relation of C.

Note that the colours of K(G) are not considered to possess absolute values in the sense of

those of WLk(G), but rather relative values. Let K ′(G) be defined similarly with respect

to ψi(C), for any i ∈ V (G). Then K ′(G) K(G), and the following corollary of Theorem

5.4.3 holds.

Corollary 5.4.4. If G has no separators of size 3k, then K ′(G), K(G) are non-

isomorphic, k-similar graphs.

And hence,

Corollary 5.4.5. If G has no separators of size 3k, WLk(K′(G)) = WLk(K(G)). Hence

if G additionally has separators of size 3(k + 1), then K(G),K ′(G) are k-equivalent.

†Note that this graph is slightly different to that obtained by a direct conversion of C, in that therelations R〉,|, where i, j /∈ E(G) are converted to non-edges.

94


This second family of k-equivalent graphs has several similarities to those of [3]. In partic-

ular, in both cases the differences between non-isomorphic pairs can be ‘shifted’ around the

graph; in the case of CFI(G) and CFI′(G) this involves ‘twisting’ an even number of (a, b)

edge pairs as described in [126]; in the case of K(G) and K ′(G) this involves applying the

ψi transformation to an even number of fibres Vi†. The basis relations R1, R2 ∈ Ri,j , for

i, j ∈ E(G), are in this way analogous to the (a, b) pairs connecting the gadgets CFI(i)

to CFI(j) in the graph CFI(G). In both cases the key property that the graphs possess is

that the separator size of the original G is proportional to the size of the k-tuples required

to distinguish the non-isomorphic pairs.

†Note that if ψ is applied to 0 (mod 2) fibres of K(G) then it preserves the isomorphism class of K(G).

95

Chapter 6

A Modified Weisfeiler-Leman

method

6.1 Graph Extensions and the k-dim WL Method

The purpose of this section is to analyse the relative properties of WLk(G) and WLk(G′),

where G′ is an extension of the graph G resulting from replacing each of the vertices of

G by some gadget, then connecting the gadgets according to a certain set of rules. This

analysis is motivated by the form of the known families of counterexample graphs, each

involving extensions of this kind applied to expander graphs.

In particular, the following theorems will be proven.

Theorem 6.1.1. If the recursive k-dim WL method succeeds in characterising the graph

G, then the recursive (k+1)-dim WL method succeeds in characterising the graph CFI(G).

Theorem 6.1.2. The recursive 1-dim WL method succeeds in characterising the graph

K(G), associated with a Klein scheme of G.

Following this, we show a more general result; namely that if a graph G is extended

to some graph G′ by replacing each vertex by an unbiased gadget of a certain type, of

which the Furer gadget [126] relating to CFI(G) is one such example, then the k-dim WL

method partitions the gadgets of G′ into the same relative colour classes as it partitions

the vertices of G. Here the colour class of a gadget refers to the sorted set of colour classes

of its constituent vertices.

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CHAPTER 6. A MODIFIED WEISFEILER-LEMAN METHOD

Definition 6.1.3. The extension of a graph G formed by replacing each vertex v ∈ V (G)

by some type of gadget h(v) will be termed unbiased if the resulting graph G′ has the

following properties:

• Whenever |d(u)| = |d(v)| for some u, v ∈ V (G), the graphs induced on h(u) and h(v)

are isomorphic.

• ∀ u, v ∈ V (G), ∃ γ ∈ Aut(G′) such that γ : h(u) 7→ h(v) if and only if

∃ ϕ ∈ Aut(G) such that ϕ : u 7→ v,

• For any x, y ∈ V (G′) such that x ∈ h(u) and y ∈ h(v) where u 6= v, we have

x, y ∈ E(G′) only if u, v ∈ E(G).

Only the first two properties are strictly necessary for the spirit of the term unbiased

to hold, however the third property is included for ease of analysis. For instance, an

alternative definition lacking the third requirement (but retaining the second) would allow

gadgets which possess mutual connections if and only if the corresponding vertices of

the initial graph are unconnected. We note that the third property holds for the sets of

counterexamples proposed in both [3] and [4].

6.1.1 Properties of the CFI graph extension

We begin with the graph extension considered in [3], defined in Section 5.3.2.

Definition 6.1.4. Let the function Λ : G → CFI(G) represent the extension of a graph

obtained by replacing all vertices with their corresponding Furer gadgets, connected as in

Section 5.3.2.

To simplify some of the later analysis, we also introduce the notation:

Definition 6.1.5. Given a vertex x ∈ V (CFI(G)), consider the function

λ−1 : V (CFI(G))→ V (G),

λ−1 : x 7→ v, ∀x ∈ CFI(x), (6.1)

which reverses the above process, mapping any vertex in the gadget CFI(v) to the vertex

v ∈ V (G).

97

6.1. GRAPH EXTENSIONS AND THE K-DIM WL METHOD

The graph CFI(G) has several important properties. Given a vertex v ∈ V (G), the pair of

vertices av,i, bv,i ∈ CFI(v), i ∈ d(v) belong to the same orbit of Aut(CFI(G)), and hence

to the same colour class in WLk(CFI(G)).

Similarly, all central vertices mS ∈M(v) of a given gadget CFI(v) also belong to the same

orbit of Aut(CFI(G)), and thus the same colour class of WLk(CFI(G)).

With the exception of the case where G is a cycle graph (this trivial case is assumed from

this point to not occur), the following further properties regarding the relative colouring

of the A,B,M vertex sets also hold.

Since the M(v) vertices are initially coloured differently to the A(v) and B(v) vertex sets,

the 2-tuples (av,x, bv,x) and (av,x, bv,y), where v, x, y ∈ V (G), x 6= y are assigned different

colours by the k-dim WL method (for k > 1).

Another simple corollary of the definition of the CFI graph extension together with the

above observations is that for all u, v ∈ V (G), u 6= v,

Sort[WL(CFI(u))] and Sort[WL(CFI(v))]

are either equal or disjoint.

Before presenting a prove of Theorem 6.1.1 we focus on a simpler, ‘warm-up’ case.

Theorem 6.1.6. Given a graph G with vertices u, v ∈ V (G), WL1(u) 6= WL1(v) if and

only if Sort[WL1(CFI(u))] 6= Sort[WL1(CFI(v))]

Proof. Consider the vertices u, v ∈ V (G) with associated gadgets CFI(u), CFI(v) ⊂CFI(G). Given a ∈ CFI(u), b ∈ CFI(v), we have

d(a) =

|d(u)| if a ∈M(u),

2|d(u)|−2 + 1 otherwise.(6.2)

Hence WL01(a) = WL0

1(b) implies that WL01(u) = WL0

1(v). Furthermore, if either a ∈M(u) and b ∈ M(v), or a /∈ M(u) and b /∈ M(v) holds, then the converse is true, and we

have WL01(a) = WL0

1(b) if and only if WL01(u) = WL0

1(v).

Conversely, assume that WLi1(a) = WLi1(b) implies that WLi1(u) = WLi1(v) for some

i ∈ Z. For any x ∈ V (G),

WLi+11 (x) = 〈WLi1(x), SortWLi1(y) : y ∈ d(x), SortWLi1(y) : y ∈ e(x) 〉, (6.3)

98


hence WLi+11 (a) = WLi+1

1 (b) implies that SortWLi1(y) : y ∈ d(a) = SortWLi1(y) : y ∈d(b), and similarly for elements of e(a) and e(b). This in turn implies that WLi+1

1 (u) =

WLi+11 (v). Hence by induction we have

WL1(u) 6= WL1(v) only if WL1(a) 6= WL1(b). (6.4)

Similarly, the converse follows if we restrict a and b such that either a ∈ M(u) and

b ∈ M(v), or a /∈ M(u) and b /∈ M(v) holds, or if we consider the sorted set of colour

classes associated with CFI(u) and CFI(v).

A similar induction proof technique can be used to show that this result also holds for k-dim

WL, for any k. A few requisite properties of the CFI graphs will first be established. Let

G be a graph, with u, v, x, y ∈ V (G). Note that WLk(u, v) = WLk(x, y) only if the number

of paths of each length connecting u, v and x, y respectively are equal [122]. Further, let

the character of a path (x1, . . . , xt) denote the ordered set of colour classes associated

with each element of the path, (WLk(x1), . . . ,WLk(xt)). Then WLk(u, v) = WLk(x, y)

only if the number of paths of each length and of each character connecting u, v and x, y

respectively are equal. Hence the following hold.

Let G be a graph, with u, v, w ∈ V (G), where u 6= v, u 6= w.

Lemma 6.1.7. Let x1, x2 ∈ CFI(u), y ∈ CFI(v). Then for k > 1, WLk(x1, x2) 6=WLk(x1, y).

Lemma 6.1.8. Let u, v ∈ E(G), u,w /∈ E(G). Let x ∈ CFI(u), y ∈ CFI(v), z ∈CFI(w). Then WLk(x, y) 6= WLk(x, z).

Corollary 6.1.9. Let x, y, z ∈ CFI(G). Then

WLk(x, y) = WLk(x, z) only if iso(λ−1(x), λ−1(y)) = iso(λ−1(x), λ−1(z)).

Proof. Firstly, recall that iso(x1 . . . xk) = iso(y1 . . . yk) if and only the relevant vertex and

edge colourings match, and

xi = xj if and only if yi = yj , and

(xi, xj) ∈ E(G) if and only if (yi, yj) ∈ E(G). (6.5)

Let λ−1(x) = u, λ−1(y) = v, λ−1(z) = w. If the pairs (u, v) and (u,w) have different initial

colours in G, then trivially the pairs of gadgets (CFI(u),CFI(v)) and (CFI(u),CFI(w))

99


have different initial colours in CFI(G). The colour of iso(u, v) further reflects which of

the following holds:

(i) u = v

(ii) u, v ∈ E(G)

(iii) u, v /∈ E(G), u 6= v.

In each case, by Lemmas 6.1.7 and 6.1.8 this information is also contained in the colouring

of WLk(x, y).

Hence a generalisation of the k = 1 result to all k can be derived.

Theorem 6.1.10. Given a graph G, where u, v ∈ V (G),

WLk(u) 6= WLk(v) only if Sort[WLk(CFI(u))] 6= Sort[WLk(CFI(v))].

Proof. In Section 6.3, for a given t-tuple z ∈ V (G)t, t < k, we define WLtk(z) recursively

by

WLtk(z) = 〈 SortWLtk((z, i)) : i ∈ V (G) 〉.

Hence by Corollary 6.1.9 above,

WL0k(u) 6= WL0

k(v) only if Sort[WLk(CFI(u))] 6= Sort[WLk(CFI(v))]. (6.6)

Let S1, S2 ⊂ V (G) be k-tuples of G, and R1, R2 ⊂ V (CFI(G)) be k-tuples of CFI(G), such

that λ−1(Ri) = Si, for i ∈ 1, 2.

Assume that for some m,n ∈ Z, WLmk (S1) 6= WLmk (S2) implies that WLnk(R1) 6=WLnk(R2), for all S1, S2 ⊂ V (G), and all such R1, R2.

Then consider a specific set of k-tuples S1, S2, R1, R2 with the property

WLmk (S1) = WLmk (S2),

WLm+1k (S1) 6= WLm+1

k (S2). (6.7)

In other words,

SortWLmk (S′1(x) : x ∈ V (G) 6= SortWLmk (S

′2(x) : x ∈ V (G),

100


where S′(x) is defined as in Eq. (5.2) to be the set of ‘neighbouring’ k-tuples to S, con-

taining x. But by the assumption above, this implies

SortWLnk(R′1(x) : x ∈ V (CFI(G)) 6= SortWLnk(R

′2(x) : x ∈ V (CFI(G)),

hence WLn+1k (R1) 6= WLn+1

k (R2).

Hence, as this assumption holds for m = 0 it holds for all m by induction.

One final preliminary is needed before addressing Theorem 6.1.1.

Lemma 6.1.11. If the recursive k-dim WL method refines the graph G to its orbits at

each step, then the recursive (k + 1)-dim WL method refines the ordered 2-tuples of G to

their orbits at each step.

Proof. This follows relatively directly from the definition of the k-dim WL method. Con-

sider the pointwise stabiliser of WLk(G), in which a single vertex belonging to a particular

colour class has been individualised. At each step of the recursive individualisation, the

k-dim WL method again refines the resulting graph to its orbits. Denote the graph result-

ing from applying the k-dim WL method, then individualising some vertex (belonging to

an orbit of size greater than one) i times by G(i)k (T ), where T = (t1, . . . , ti), tj denoting

the vertex individualised at step j in the recursive method.

Given two ordered pairs (u, v), (x, y) ∈ V × V , assume that

(WLk(u),WLk(v)) = (WLk(x),WLk(y)), and

WLk(v)|G

(1)k (u)

= WLk(y)|G

(1)k (x)

. (6.8)

Denote by Sk(a, b) the set of (k)-tuples containing a and b, Sk(a) = R ∈ V (G)k : a, b ∈ R.

Then since G(1)k (a) is refined to its orbits by the k-dim WL method, Eqs. (6.8) above imply

that

(WLk+1(u),WLk+1(v)) = (WLk+1(x),WLk+1(y)), and

WLk+1(Sk+1(u, v)) = WLk+1(Sk+1(x, y), (6.9)

and hence that WLk+1((u, v)) = WLk+1((x, y)). Hence the colour of ordered 2-tuples in

WLk+1 is equal only if they belong to the same orbit of V × V in G.

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Combining the results of Theorem 6.1.10, Lemma 6.1.11 and the observations regarding

the properties of WLk(CFI(G)) at the start of this subsection, we can now return to

Theorem 6.1.1 introduced at the beginning of Section 6.1.

Theorem 6.1.12. If the recursive k-dim WL method succeeds in characterising the graph

G, then the recursive (k+1)-dim WL method succeeds in characterising the graph CFI(G).

Proof. Assume ∃u, v ∈ V (G), such that WLk(u) 6= WLk(v). Then by Theorem 6.1.10

for any r, s ∈ V (CFI(G)), λ−1(r) = u, λ−1(s) = v, it follows that WLk(r) 6= WLk(s).

Furthermore the graph induced on any given gadget of CFI(G) is itself refined to its

orbits (in that the A/B subsets are separated from the M subset in non-trivial cases).

For a given v ∈ V (G), the orbits of the set of central vertices M(v) ⊂ CFI(v) depend

only on the orbit of the vertex v in Aut(G). Hence the central M vertices of CFI(G)

are refined to their orbits by the k-dim WL method, and hence by the (k + 1)-dim WL

method. However the A and B vertices of each gadget CFI(v) correspond to di-edges

of G, in that the orbit of a particular vertex au,v depends on the orbit of the ordered

2-tuple (u, v) in Aut(G2). Explicitly, ∃φ ∈ Aut(CFI(G)), φ : au,v 7→ ax,y if and only if

∃ γ ∈ Aut(G2), γ : (u, v) 7→ (x, y).

Now as the set of colour classes from the WL method is a graph invariant, and so cannot

refine further than the orbit partition, it follows from the previous argument that the

n-dim WL method refines CFI(G) to its orbits if it refines the ordered 2-tuples of G to

their orbits. Hence the result follows from Lemma 6.1.11.

Corollary 6.1.13. If G can be characterised by the recursive k-dim WL method, then the

graph Λi(G), in which Λ is applied i times to G, can be characterised by the (k + 1)-dim

WL method.

Proof. Follows from the proof of 6.1.1. In particular, note that applying Λ i times to

G still results in a graph in which vertices represent at worst di-edges of G, in that

the automorphism group of Λi(G) does not involve automorphisms of t-tuples of G for

t > 2.

Note that whereas the recursive k-dim WL method is sufficient to characterise G, the

(k+ 1)-dim WL method is required to characterise the extension CFI(G). This is a direct

result of there being vertices in CFI(G) that directly represent di-edges, or ordered pairs

of vertices, in G. Similarly, if a graph extension was constructed that contained vertices

102


representing 3-tuples of G, the (k + 2)-dim WL method would be required in a proof

proceeding as above. There are some fairly contrived possibilities for getting around this

requirement, for instance by altering the recursive WL method, restricting the vertex

individualised at each step to belong to the set Mu for some u ∈ V (G). Since these central

vertices ‘encode’ only a single vertex in G, unlike the A and B sets they will necessarily be

refined to their relative orbits by the k-dim WL method. Alternatively, the recursive WL

method (when acting on a graph of the type CFI(G)) could be restricted to individualising

an entire gadget at each step. These alternative method require foreknowledge of the graph

type of interest however, and as such are of less interest to a discussion on possible general

graph isomorphism algorithms.

Requiring an extension to the (k + 1)-dim method may seem prohibitive from an effi-

ciency viewpoint, in that extensions similar to those in [3] and [4], in which vertices are

present whose orbit depends on the orbit of some t-tuple in the original graph, for t > 2,

might easily be produced. A method involving such t-tuples in an unbiased way, in the

sense of definition 6.1.3, might be expected to develop alternative weaknesses with grow-

ing t however. For the moment such potential extensions are beyond the scope of this

work, however they do represent a potentially promising direction in which to look for

k-equivalent graphs for which the arguments of this section do not apply.

6.1.2 K(G) Counterexamples

Here we consider the k-equivalent pairs K(G) and K ′(G), given a cubic graph G. The

analysis of these counterexamples is greatly simplified due to the following properties:

Let C = (V,R) be the coherent configuration associated with K(G), and Aut(C) be the

automorphism group of C, where Aut(Ci,j) is the automorphism group of the induced

coherent configuration on Vi × Vj . Then, from the results of [4] (Lemma 5.3),

|Aut(Ci,j)| =

4 if i = j

8 if (i, j) ∈ E(G)

16 otherwise.

(6.10)

In particular, note that no fibres span more than one Vi, and the orbits of V are simply

the sets Vi, ∀i ∈ V (G). Further, let C∗ be the coherent configuration resulting from

individualising a single point of C, where K(G)∗ is the associated graph.

103


Corollary 6.1.14. |Aut(C∗)| = 1; no non-trivial automorphisms exist for C∗.

Consider the following adaption of the 1-dim WL method, accepting an edge-coloured

graph as input. Denote the colour of the edge (u, v) ∈ E(K(G)) by C(u, v), and let

WLt1(u) = 〈WLt−11 (u), Sort(WLt−1

1 (v),C(u, v)) : v ∈ d(u),

Sort(WLt−11 (v) : v ∈ e(u) 〉. (6.11)

Theorem 6.1.2 follows immediately.

Theorem 6.1.2. The recursive 1-dim WL method succeeds in characterising the graph

K(G), associated with a Klein scheme of G.

Proof. Let u, v ∈ V (K(G)), where u ∈ Vi, v ∈ Vj for some i 6= j. If follows that u and v

have incident edge-colour sets that do not coincide. Then WL11(u) 6= WL1

1(v). Hence the

1-dim WL method described above refines K(G) to its orbits.

Let u ∈ Vi be the vertex of K(G) individualised in K(G)∗. The remainder of Vi are

connected to u each via edges of a different colour, hence for any x, y ∈ Vi, WL11 assigns

different colours to x, y. Similarly, consider Vj such that i and j are at distance n in G.

Then WLn1 assigns different colours to each vertex of Vj . Hence K(G)∗ is refined to its

orbits (the discrete partition) by WL1.

The ease of proving this result compared to the corresponding result regarding the CFI

counterexamples stems from each ‘gadget’ in K(G) being essentially assigned a unique

colour (explicitly in [4], implicitly here via the the unique colouring of each di-edge of G).

This distinction between di-edge colouring could be removed, with the Ri,j basis relations

of Section 5.4 merged, such that the sets of basis relations

Sx =⋃

i,j:(i,j)∈E(G)

Rx, x ∈ 1, 2, and (6.12)

T =⋃

i,j:(i,j)/∈E(G)

Ri,j (6.13)

are each merged into a single basis relation, forming three distinct subsets of C †, and with

the basis relations of all individual Vi being merged into three basis relations similarly.

†Explicitly, leaving the three sets of relations; S1, S2 and T .

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We will state without proving the following proposition (which follows from the results

of [4]).

Proposition 6.1.15. Merging the basis relations as detailed above preserves the properties

of k-similarity and non-isomorphism between C and C′.

Indeed, the automorphism group of the resulting coherent configuration has orbits with

the following properties. Vertices u, v are in the same orbit if they belong to a single Vi.

The vertices of Vi and Vj belong to the same orbit if and only if i, j are in the same orbit

of G.

Hence the question of whether WLk refines K(G) to its orbits reduces to a similar problem

as that of the previous section. A similar proof can be constructed, with one important

distinction. Since the vertices of each Vi correspond to a single vertex of G, rather than a

2-tuple of G as for CFI(G), the following result is obtained.

Proposition 6.1.16. If the k-dim WL method succeeds in refining G to its orbits, then

it succeeds in characterising K(G) to its orbits.

The proof is along the same lines as that of the previous section, although considerably

simpler due to the above observation.

6.1.3 General graph extensions

Recall the generalised graph extension G → G′ defined in 6.1.3 (relative to an ‘unbiased’

gadget) at the beginning of this section.

Definition 6.1.3. The extension of a graph G by replacing each vertex v ∈ V (G) by some

type of gadget h(v) will be termed unbiased if the resulting graph G′ has the following

properties:

• Whenever |d(u)| = |d(v)| for some u, v ∈ G, the graphs induced on h(u) and h(v)

are isomorphic.

• ∀u, v ∈ V (G),∃ γ ∈ Aut(G′), γ : h(u) 7→ h(v) iff ∃ϕ ∈ Aut(G), ϕ : u 7→ v,

• For any x, y ∈ V (G′) such that x ∈ h(u) and y ∈ h(v), we have x, y ∈ E(G′) only

if u, v ∈ E(G).

105

6.2. ORBIT CASE

Further to this definition of an unbiased gadget, the graph extensions considered will be

assumed to have the following property relative to the k-dim WL method.

We assume that pairs of gadgets corresponding to neighbouring vertices of G are dis-

tinguished from those corresponding to non-edges of G. Note that the graph extensions

defined in [3] and [4] satisfy this property.

This assumption implies that the colour of a k-tuple in G′ depends on the isomorphism

class of the corresponding k-tuple in G. Hence by induction, as in Section 6.1.1, the

colour of a k-tuple in G′ also depends on the colour class (resulting from the k-dim WL

method) of the corresponding k-tuple in G. Hence this implies that the k-dim WL method

partitions the gadgets of G′ into the same relative colour classes as it partitions the vertices

of G.

6.2 Orbit Case

Up to this point we have been interpreting the results of [3, 4] as demonstrating that the

k-equivalent CFI pairs cannot be directly distinguished, in that any method of producing

a canonical certificate from each graph using the colour sets resulting from the k-dim WL

method will yield identical certificates. However we argue that this does not imply that

the WL method (or some variant of this method) cannot be used indirectly to solve GI. In

particular, recall that among the problems polynomial-time equivalent to the GI problem

is that of determining the orbits of the automorphism group of a given graph. A method

that can partition the vertex set of any graph down to its orbits can trivially solve GI by

simply refining the graph to its orbits, stabilising some vertex from a given orbit, then

accepting the resulting graph as input and repeating until the discrete partition is reached.

At this point an explicit non-trivial automorphism of the graph has been found (if any such

exist), and the method can be repeated to find a set of generators for the automorphism

group. Alternatively, a method that can partition the vertex set of any graph down to

its orbits can simply be directly applied to the union of two graphs to determine if they

belong to the same isomorphism class.

Hence the focus of the following sections will be purely on the equitable partitions resulting

from the WL method, rather than the related graph invariant.

106


6.2.1 Counterexamples (orbit case)

If the goal of the WL method is instead considered to be refining the vertex set of a graph

down to its orbits, the set of known counterexamples differs, since graphs that cannot be

directly distinguished by comparing certificates could still be indirectly distinguished by

a recursive WL method, provided that at each step of the recursion the orbits could be

successfully found.

Such a recursive k-dim WL method would proceed as follows. Apply the k-dim WL

method to the graph until the equitable partition is reached, at which point this partition

is assumed to be an ordered set of orbits of the graph. Without loss of generality, any

vertex from the lexicographically smallest (for example) orbit is then stabilised, and this

process is repeated until the discrete partition is reached. If at each step the equitable

partition corresponds to the orbit set, a canonical ordering of the vertex set is obtained,

characterising the graph. Alternatively, by choosing different sets of representative vertices

to stabilise at each step, generators for the automorphism group can be efficiently obtained.

The success of this procedure is of course dependent on the ability to discover the orbits at

each step. However, we briefly note that this method may succeed for graphs where a direct

application of the WL method fails, and additionally that both success and failure of this

method occur in polynomial time (together with the knowledge of which has occurred†).

Now the results of [3, 4] demonstrate that the k-equivalent pairs described cannot be dis-

tinguished directly, and hence that, for instance, the union of such a pair cannot itself

be partitioned down to its orbits using the k-dim WL method. In this trivial case where

the graph under consideration is simply the union of two k-equivalent graphs, say CFI(G)

and CFI′(G) for some connected graph G, this inability to determine the orbits of the

combined graph H = CFI(G)∪CFI′(G) directly can be easily circumvented, under certain

assumptions. Provided the original graph G can be refined to its orbits using the k-dim

WL method, and further that this refinement can be recursively applied (with pointwise

stabilisation at each step) to completely characterise the graph (calculating its automor-

phism group), then in Section 6.1 we show that CFI(G) and CFI′(G) can also be separately

characterised by recursive application of the (k+1)-dim WL method. Then since H can be

trivially decomposed into its k-equivalent sub-constituents (by separating the disconnected

components of either H or H, depending on the definition of the ‘union’ of graphs), we can

†A simple polynomial time extension to the method can be used to determine generators for the completeautomorphism group, which in turn can be efficiently verified.

107

6.2. ORBIT CASE

apply the recursive WL method to each of the two k-equivalent subgraphs individually,

determine that these are non-isomorphic graphs, and hence characterise the composite

graph H.

The above example applies a method for decomposing a graph for which the WL method

has been proven to fail into its k-equivalent subgraphs. Of course, such a method cannot be

so easily applied in general. Before considering how a generalised decomposition method

might proceed, we will consider precisely what types of graphs have been found for which

the WL method has been proven (in the work of [3, 4]) to not determine the orbits. The

union of two or more k-equivalent graphs as considered above is one such graph, albeit a

trivial case.

Consider a graph H that can be directly deduced from the results of [3, 4] to not be

partitioned down to its orbits by the k-dim WL method. Then trivially, H must contain

at least two mutually k-equivalent subgraphs, S1 and S2. Any differences in the way

these subgraphs are connected to the remainder of the graph, relative to the colour classes

resulting from the k-dim WL method, may distinguish them (or at least have not been

proven not to do so), hence assume no such differences exist. Furthermore, consider a

single orbit of a given k-equivalent graph S. Any difference in the connections of the

elements of this orbit to the rest of the graph may yield enough information regarding the

internal structure of S such that its property of k-equivalence is destroyed. Hence such

differences will also be assumed to not exist†.

If we consider the orbit partition of H, in which each cell of the partition is a distinct

orbit of Aut(H), then such a subgraph S, essentially a generalisation of a module of a

graph, will be termed to be connected cell-wise symmetrically (CWS) to the remainder of

the graph (relative to the orbit partition in this case), defined as follows.

Definition 6.2.1. Consider an ordered partition π(G) = (V1, V2, . . . , Vr) of the graph G

into cells (or colour classes), and define θ : V (G)→ π(G) to determine the cell of a given

vertex v ∈ V (G). Then a subgraph S ⊂ G is connected cell-wise symmetrically (or CWS)

within G, alternatively termed a cell-wise symmetric (or CWS) subgraph of G, relative to

π(G), if:

∀ u, v ∈ V (S) such that θ(u) = θ(v), we have d(u)|(G\S) = d(v)|(G\S). (6.14)

†Note that in this work we are only concerned with graphs for which the recursive k-dim WL methodhas been proven to fail; we wish to know exactly what counterexamples can be directly constructed fromthe results of [3, 4], hence such differences cannot be allowed without a further extension to this work.

108


In other words, any elements of S in the same cell of π(G) have identical neighbour sets

outside S (in G\S).

Definition 6.2.2. A subgraph S ⊆ G will be termed prime if it has no proper, non-trivial

CWS subgraphs, and is itself a non-trivial CWS subgraphs of G. This is defined implicitly

with respect to the k-dim WL method.

Example 6.2.3. Let G be the n-cube, in which the vertices are associated with the related

2n points in Zn2 . Associate with G the partitioning π(G) = (x, V (G)\x), where

x ∈ V (G) is the vertex with associated bit-string (0 . . . 0). This would be for instance

the orbit set resulting from individualising the vertex x. Then the subset Sc ⊂ V (G),

Sc = v ∈ V (G) : dist(v, x) = c, corresponding to the set of points with fixed Hamming

weight c, is a CWS partition of G relative to π(G).

For the remainder of this work, we will consider CWS subgraphs to be defined relative to

either the orbit partition or the colour classes resulting from the k-dim WL method. In the

former case such subsets will be referred to as orbit-wise symmetric (OWS) subsets, and

in the latter case simply as CWS subsets, with the ‘relative to the colour classes resulting

from the k-dim WL method’ specifier dropped for the purposes of brevity.

The notion of CWS subgraphs of a graph can be extended to relative connections between

mutually k-equivalent subgraphs of a graph. In particular, the properties of known coun-

terexample graphs discussed above refer to differences in the relative connections between

non-isomorphic k-equivalent subgraphs and the remainder of the graph. Before formalis-

ing this concept into a definition of mutual CWS subsets, it will be instructive to consider

a simpler case.

In particular, let R and S be isomorphic, vertex-disjoint, mutually k-equivalent subgraphs

of a graph G. As R and S are mutually k-equivalent, ∃ θ : V (R)→ V (S) such that

WLk(u) |R= WLk(θ(u)) |S , ∀ u ∈ V (R). (6.15)

We assume that R and S are connected CWS within G, relative to the WLk colour classes

corresponding to their respective induced graphs, such that for all u, v ∈ V (R) such that

WLk(u) |R= WLk(v) |R, we have

WLk(u) |G= WLk(v) |G, (6.16)

109

6.2. ORBIT CASE

and similarly for S. Further, R and S will have the property that their cell-wise connections

to G\R and G\S respectively are equivalent, in the sense that

WLk(d(u)) |G= WLk(d(θ(u))) |G, ∀ u ∈ V (R), (6.17)

which in turn implies that

WLk(u) |G= WLk(θ(u)) |G, ∀ u ∈ V (R). (6.18)

Note that if Eq. (6.18) holds for one such mapping θ defined as in Eq. (6.15), it holds for

all such mappings. In other words, in the terminology of Section 5.4, there exists a weak

k-automorphism of G that maps R to S, in that

WLk(R) |G= WLk(S) |G . (6.19)

Finally, we assume that R and S are the only mutually m-equivalent graphs, m ≥ k, for

which the above holds.

We argue that in this situation, either there exists an automorphism of G that maps R to

S, or G represents a novel type of k-equivalent graph, the existence of which is currently

unknown.

Firstly, note that the above properties are trivially consistent with the case where an

automorphism of G maps R to S, and that in either case G must be k-equivalent, in that

replacing S with a non-isomorphic mutually k-equivalent copy of S† while retaining the

mapping θ results in a graph which is mutually k-equivalent to G.

In the situation where no such automorphism exists, we note that the graph G′ obtained

by replacing R and S each by a single vertex‡ (vRandvS respectively), retaining the rep-

resentative CWS connections outside R and S respectively, is also k-equivalent, in that

WLk(vR) |G′= WLk(vS) |G′ ,

however no automorphism of G′ maps vR to vS . Hence we are essentially shifting the

k-equivalence property of G outside R and S.

Now the arguments directly prior to Definition 6.2.1 essentially state that known classes

†Which exists by the assumption that S is k-equivalent.‡This process is explicitly defined in Definition 6.4.8.

110


of graphs which are not partitioned to their orbits by the k-dim WL method must contain

a pair of non-isomorphic graphs with the properties ascribed to R and S above. However

since we have assumed that no further such mutually m-equivalent graphs exist in G, for

m ≥ k, this does not occur, and hence the graph G represents a novel type of k-equivalence.

The above discussion serves to inform the following definition of mutually CWS subsets of

a graph. Let S = A1, A2, . . . , Ai be a set of vertex-disjoint, CWS subgraphs of a graph

G, such that the elements of S are all pairwise k-similar. The graphs Ax, where x ∈ [i],

are said to be mutually CWS if they have k-equivalent connections in G in the following

sense.

Definition 6.2.4. Consider the graph G′ in which all pairs of non-isomorphic mutually

m-equivalent subgraphs of G are replaced by isomorphic mutually m-equivalent graphs of

the same m-equivalence class, for all m ≥ k. In the case where the subgraphs Ax are not

k-equivalent, S is unchanged. It follows that G′ ∼k G, however all mutually m-equivalent

subgraphs of G′ now belong to a single isomorphism class. Then elements of S are said to

be mutually CWS (or alternatively said to be connected CWS to each other) within G, if

for all Ax, Ay ∈ S, there exists a φ ∈ Aut(G′) such that φ(Ax) = Ay (and hence S is in

this sense vertex-transitive).

Note that the definition of mutually CWS subgraphs corresponds to a specific value of k,

which will be clear from the context where not explicitly stated.

Corollary 6.2.5. If R and S are mutually CWS subgraphs of G relative to the k-dim WL

method, then R ∼k S.

Example 6.2.6. Let S = A1, A2, . . . , Ai be a set of vertex-disjoint, CWS, mutually

k-equivalent subgraphs of a graph G. Then for all Ax, Ay ∈ S, ∃ θx,y : V (Ax) → V (Ay),

such that ∀u ∈ V (Ax),

WLk(u) |Ax= WLk(θ(u)) |Ay . (6.20)

In particular, for all x, y ∈ [i] and u ∈ V (Ax),

Sort[WLk(d(u)) |(G\S)] = Sort[WLk(d(θx,y(u))) |(G\S)], and

∃Az ∈ S s.t. Sort[WLk(d(u)) |Ay ] = Sort[WLk(d(θx,z(u))) |Ax ].

111

6.2. ORBIT CASE

Example 6.2.7. A set S = A1, . . . , Ai of CWS subgraphs of G are trivially mutually

CWS if the following hold

(i) The Ax ∈ S are all pairwise mutually k-equivalent.

(ii) For all x, y ∈ [i] and u ∈ Vx, v ∈ Vy such that u ∼k v, we have

d(u) ∩ (G\Vx) = d(v) ∩ (G\Vy). (6.21)

Note that a graph G containing a CWS k-equivalent subgraph S is not necessarily itself

k-equivalent, in that although the connections between S and G are CWS, they are not

necessarily OWS. However several trivial cases, such as a CWS k-equivalent subgraph S

in which all vertices have identical neighbour sets outside of S, can be constructed in

which the resulting graph G is provably also k-equivalent. Hence in addressing the known

k-equivalent graphs, we make the following assumption.

Assumption 6.2.8. Any graph G that contains a CWS k-equivalent subgraph is itself

k-equivalent.

Whilst graphs exist for which this does not hold (in fact several were readily found in the

course of this work), this assumption makes the following task of addressing all known

counterexample graphs more difficult (in the sense that additional graphs must be consid-

ered), and so can be made without weakening the end results.

Following the preceding set of definitions, and the properties of known counterexample

graphs discussed above, we now have the notation required to define the following family

of graphs.

Definition 6.2.9. The family Mk of graphs is defined as including those graphs for which

the k-dim WL method has been shown to fail to partition the vertex set down to its orbits.

and not including graphs for which the (k+ 1)-dim WL method has been shown to fail in

this sense. All graphs G ∈Mk have the following properties:

1. G and G are connected.

2. G contains two non-isomorphic, mutually CWS subgraphs S1, S2, each connected

CWS to G relative to the colour classes of WLk(Si) |Si .

3. G contains no pair of non-isomorphic, mutually CWS, m-equivalent subgraphs,

where m > k.

112


Note that if (2) does not hold, and such a mutually CWS pair of subgraphs does not exist,

then the k-dim WL method will not have been proven to fail to provide the orbits of G.

Similarly if (3) does not hold, G ∈Mk+1, hence G /∈Mk.

Note that the set of graphs Mk is not intended as a complete set of graphs that the k-

dim WL method fails to refine down to orbits, but instead as including the set of graphs

for which this has been previously proven to occur. For instance, it may be possible to

construct a graph not containing mutually CWS k-equivalent graphs that still cannot be

successfully characterised by the recursive k-dim WL method. However such graphs have

not been proven to exist, and the primary object of this work is simply to discuss the

possibility that a variant of the WL method might be used to solve GI, not to prove that

it actually can.

Note that the set Mk contains those graphs for which the k-dim WL method has been

proven to fail, in the sense that it cannot partition the vertex set of such a graph down

to its orbits. In Section 6.4, we detail an algorithm employing the recursive WL method

that can characterise these graphs, under certain assumptions, by first applying a decom-

position method that isolates the relevant k-equivalent subgraphs. These subgraphs are

then individually characterised using the standard recursive WL method, at which point

the non-isomorphic k-equivalent subgraphs are distinguished, and the original graph can

be characterised.

6.3 Properties of the WL method

Before continuing our discussion regarding counterexample graphs, we will first take a brief

interlude to establish some basic properties of the k-dim WL method. These properties will

prove useful in constructing some of the proofs of Section 6.4. In particular, the relation

between the colour classes of WLk(G) and the CWS subgraphs of G will be explored.

Firstly, the CWS closure cl(S) of a subgraph S ⊂ G is defined as the smallest CWS subset

of G containing S.

Consider the following method for constructing the CWS closure cl(u, v) of a pair of

vertices u, v ∈ V (G) belonging to the colour class c, such that u∼kv in G, where WLk(u) =

c.

(i) Begin with S0 = u, v.113

6.3. PROPERTIES OF THE WL METHOD

(ii) For each x, y ∈ Si such that WLk(x) = WLk(y), find the set

R(x, y) := (d(x) ∩ e(y)) ∪ (e(x) ∩ d(y)).

(iii) Set Si+1 = (⋃x,y R(x, y)) ∪ Si.

(iv) When St is equitable, such that St = St+1, set cl(u, v) = St.

Lemma 6.3.1. Let WLk(G) contain colour classes c1 and c2. If there exists a vertex

v ∈ [c2] such that v ∈ cl([c1]), then [c2] ⊂ cl([c1]).

Proof. Let S0 = x1, x2, for x1, x2 ∈ [c1]. If there exists a u ∈ [c3] for some colour class

c3 ∈WLk(G), such that u ∈ S1, then either

u ∈ d(x1), u /∈ d(x2), or

u /∈ d(x1), u ∈ d(x2).

Since c3 is a colour class of WLk(G), then for all such v ∈ [c3],

∃ xi, xj ∈ [c1] s.t. v ∈ d(xi), v /∈ d(xj),

hence [c3] ⊂ cl([c2]).

Similarly, if there exists a v ∈ [c2] such that v ∈ Si, where Si−1 ∩ [c2] = ∅, then one (or

more) of the following hold:

• ∃ xi, xj ∈ [c1], xi, xj ⊂ Si−1, s.t. v ∈ d(xi), v /∈ d(xj).

• ∃ c3 ∈WLk(G), y1, y2 ∈ [c3], yi, yj ⊂ Si−1, s.t. v ∈ d(yi), v /∈ d(yj).

In the former case, [c2] ⊂ cl([c1]) as above. In the latter case, [c2] ⊂ cl([c3]), where

[c3] ⊂ cl([c1]) in turn, hence [c2] ⊂ cl([c1]).

Corollary 6.3.2 (‘No One-Way Closure’). [c2] ⊂ cl([c1]) if and only if [c1] ⊂ cl([c2]).

Proof. Follows immediately from the proof of Lemma 6.3.1.

This ‘no one-way closure’ result only applies when considering the closure of entire colour

classes. If instead considering subsets S1 ⊂ c1, S2 ⊂ c2, then trivially we can have

S1 ⊂ cl([S2]) and S2 * cl([S1]).

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A note on notation : In the remainder of this section we consider the colour class

assigned to ordered t-tuples by the k-dim WL method, for varying t. In denoting

the colour class of an some ordered t-tuple (x1, x2, . . . , xt), we will use the notation

WLk(x1, x2, . . . , xt) to refer to the more explicit WLk((x1, x2, . . . , xt)), with the addi-

tional brackets dropped for aesthetic reasons. When considering unordered t-tuples, the

delimiter WLk(x1, x2, . . . , xt) will always be explicitly used.

The following result relates to the conversion between the colour classes of k-tuples result-

ing from the k-dim WL method, and the associated colouring of t-tuples, for t < k.

Theorem 6.3.3. Let S1 = (x1, . . . , xk) and S2 = (y1, . . . , yk) be ordered k-tuples of V (G).

Then WLk(S1) = WLk(S2) in G only if WLk(xa1 , . . . , xat) = WLk(ya1 , . . . , yat) in G for

all ai ∈ [k], t < k.

Proof. The colouring of t-tuples stemming from the k-tuple colouring is defined to be

calculated such that t-tuples have the same colour if and only if there are no differences

between the associated k-tuple colour classes that could possibly distinguish them. This

definition is far from explicit however, hence in what follows we will consider various

possible t-tuple colourings that satisfy this condition, with the aim being to settle on the

simplest possible such colouring system.

To satisfy the above condition, a colouring of (k − 1)-tuples need only encompass the

information contained in

WLk(x1, . . . , xk−1) = 〈 SortWLk(x1, . . . , xk−1, i) : i ∈ V (G) 〉.

However (k − 2)-tuples satisfy the above condition, and are hence k-similar if and only

if more complicated sets of k-tuple colourings are equal, involving ordered, nested sets of

k-tuple and (k − 1)-tuple colourings. Simplifying the characterisation of t-tuple colouring

will hence be potentially quite useful.

Recall from Eq. (5.5) that

WLk(x1, . . . , xk) = 〈 Sort(WLk(x1, . . . , xk−1, i), . . . ,WLk(i, x2, . . . , xk)) : i ∈ V (G) 〉,(6.22)

hence the theorem holds directly for t = k − 1.

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For t = k − 2, a system of colouring satisfies the above condition only if (k − 2)-tuple x

will have colours corresponding to some ordered, nested list such as:

WLk(x) = 〈 Sort(WLk(x, i),

Sort(WLk(x, i, j),WLk(x, j), . . .) : j ∈ V (G) : i ∈ V (G) 〉, (6.23)

where the unspecified continuation involves further k- and (k − 1) tuples involving k − 3

elements of (x) together with i and j. However by Eq. (6.22) k-tuples have equal colourings

only if their corresponding ordered sets of neighbouring (k−1)-tuples have equal colourings,

hence Eq. (6.23) can be simplified to:

WLk(x) = 〈 SortSortWLk(x, i, j) : j ∈ V (G) : i ∈ V (G) 〉

= 〈 SortWLk(x, i) : i ∈ V (G) 〉, (6.24)

hence the theorem holds for t = k − 2 also.

Similarly, for general t the factors of a given t-tuple involving (t+ i)-tuples, where t+ i < k

can be incorporated into the relevant k-tuple factors.

Hence the t-tuple x = (x1, . . . , xt) can be consistently coloured by:

WLk(x) = 〈 Sort. . . SortWLk(x, i1, . . . , ik−t) : ik−t ∈ V (G) . . . : i1 ∈ V (G) 〉

= 〈 SortWLk(x, i) : i ∈ V (G) 〉, (6.25)

without losing any relevant information present in the k-tuples (meaning all information

that could potentially distinguish between t-tuples is all included).

One important implication of this result is that the process of converting the k-tuple

colourings to t-tuple colourings (for any t < k) and then back to k-tuple colourings cannot

partition the set of k-tuples further. Also note the following immediate corollary.

Corollary 6.3.4. For ordered t-tuples x = (x1, . . . , xt) and y = (y1, . . . , yt) of G,

WLk(x) = WLk(y) in G only if WLk(xa1 , . . . , xar) = WLk(ya1 , . . . , yar) in G for all

such r-tuples, r < t, in which ai ∈ [k] ∀ i ∈ [r].

This result also applies for (k + i)-tuples, where i ≥ 1, for which a similar simplified

recursive definition can be constructed. In particular, let S1 = (x1, . . . , xk) and S2 =

(y1, . . . , yk) be ordered k-tuples of V (G), let x be an ordered i-tuple of V (G), and let

116


(S1,x), (S2,x) be the ordered (k+ i)-tuples resulting from concatenating the respective k

and i tuples. Then the following result holds.

Theorem 6.3.5. WLk(S1) = WLk(S2) only if

SortWLk(S1,x) : x ∈ V (G)i = SortWLk(S2,x) : x ∈ V (G)i, for all i > 0.

Proof. Given some (t+1)-tuple z = (z1, . . . , zt+1), denote by z′ the ordered set of associated

t-tuples contained in z, such that z′ = ((z1, . . . , zt), . . . , (z2, . . . , zt+1)). Similarly to the

proof of Theorem 6.3.3 we will define the colouring of (k + i)-tuples such that there two

(k+i)-tuples have the same colour if and only if no differences exist between the associated

k-tuples that could possibly distinguish them.

Hence the colouring of a (k+ i)-tuple z = (z1, . . . , zk+i) can be constructed recursively by:

WLk(z) = 〈WLk(z′)〉 (6.26)

Then by definition the theorem holds for i = 1. Assume that it also holds for i = t, and

let z as defined above be a (k+ t+ 1)-tuple of V (G). Then by Eq. (6.26) it also holds for

i = t+ 1, hence by induction it holds for all i > 0.

Combining the preceding two theorems, regarding t-tuples where t < k and t > k respec-

tively, we obtain the following corollary.

Corollary 6.3.6. Let x and y be t-tuples of V (G), for some t < k. Then WLk(x) =

WLk(y) only if the corresponding sorted sets of (t+ i)-tuple colours are also equal, for all

i > 0.

Proof. Follows directly from Theorems 6.3.3 and 6.3.5.

Note that the colour class of a k-tuple resulting from the k-dim WL method are linked

to the paths of each length connecting elements of the k-tuple. In particular, Alzaga et

al. [122] show that k-tuples x = (x1, . . . , xk) and y = (y1, . . . , yk) have the same colour

only if for any i, j ∈ [k] and m ∈ Z, the number of paths of length z connecting xi to xj ,

and yi to yj respectively are equal. This extends trivially to the case where the number of

paths of each character are considered, where the character of a path denotes the ordered

set of colour classes associated with each element of the path in turn.

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It will also be useful to establish a relationship between the colour class c of a vertex

v ∈ V (G), and the properties of the set of CWS closures cl(v, v′) : v′ ∈ [c].

Theorem 6.3.7. Let x1, x2, y1, y2 ∈ V (G). If WLk(x1, x2) = WLk(y1, y2), for k ≥ 3, then

the sets A = cl(x1, x2) and B = cl(y1, y2) have the following properties.

(i) |A| = |B|,

(ii) For all ci ∈WLk(G), |v ∈ A : WLk(v) = ci| = |v ∈ B : WLk(v) = ci|,

(iii) A is prime if and only if B is prime, and

(iv) The graphs induced on A and B respectively are k-similar.

Proof. Recall the notation regarding cl(x1, x2), where S0 = x1, x2, R(x, y) = (d(x) ∩e(y)) ∪ (e(x) ∩ d(y)) and Si+1 = (

⋃x,y R(x, y)) ∪ Si.

For u, v ∈ V (G), let u ∈ S1, and v /∈ S1. Then iso(x1, x2, u) 6= iso(x1, x2, v) and hence

WLk(x1, x2, u) 6= WLk(x1, x2, v).

Further, let u ∈ St+1 such that u /∈ St, and let v /∈ St. Then

∃ i, j ∈ St such that u ∈ R(i, j), and

@ i, j ∈ St such that v ∈ R(i, j). (6.27)

Assume that for all l ∈ St,m /∈ St, we have WLk(x1, x2, l) 6= WLk(x1, x2,m). Note that

if k ≥ 3 we then have WLk(x1, x2, u) = WLk(x1, x2, v) only if there exist some l′ ∈ St and

m′ /∈ St such that

WLk(l, u) = WLk(l′, v), and

WLk(m,u) = WLk(m′, v). (6.28)

Hence u ∈ St+1 such that u /∈ St, and v /∈ St implies that WLk(x1, x2, u) 6= WLk(x1, x2, v).

Since u ∈ S1, and v /∈ S1 implies that WLk(x1, x2, u) 6= WLk(x1, x2, v), then by induction

this holds for all t > 1 also.

Furthermore, by Corollary 6.3.4, for all u, v ∈ St, WLk(x1, x2, u) = WLk(x1, x2, v) only if

WLk(u) = WLk(v), hence (ii) holds (and as a corollary, (i) holds).

Let i, j ∈ A, WLk(i) = WLk(j), such that cl(i, j) 6= A (i.e. A is not prime). Then there

exists a vertex u ∈ A, u /∈ cl(i, j). Assume further that B is prime, and so @ l,m ∈ B,

118


WLk(l) = WLk(m) such that v /∈ cl(l,m) for some v ∈ B. Hence @ l,m ∈ B such that

WLk(l,m) = WLk(i, j), from which it follows (from Corollary 6.3.4 and the proof of (ii))

that WLk(x1, x2) 6= WLk(y1, y2). Hence (iii) holds.

Similarly, let z be a t-tuple of V (G). By Corollary 6.3.6, (x1, x2) and (y1, y2) belong to

the same colour class only if

SortWLk(x1, x2, z) : z ∈ V (G)t = SortWLk(y1, y2, z) : z ∈ V (G)t,

for all t > 0. In particular, note that for any z1 ∈ A and z2 /∈ A, we have

WLk(x1, x2, z1) 6= WLk(x1, x2, z2).

Hence

SortWLk(x1, x2, z) : z ∈ A = SortWLk(y1, y2, z) : z ∈ B,

and in particular,

WLk(A) = WLk(B).

Finally, note that if two CWS subgraphs of G are k-similar within G, then the respective

induced graphs are also k-similar, from which (iv) follows.

Definition 6.3.8. Let v ∈ [c], such that c = WLk(v) |G. Denote the CWS spectrum of a

vertex v to be the set of pairwise CWS closures Cv = cl(v, v′) : v′ ∈ [c].

Then the following corollary of Theorem 6.3.7 holds.

Corollary 6.3.9. Let u, v ∈ V (G). Then WLk(u) = WLk(v) only if there is a matching

between elements of Cv and Cu in the sense of Theorem 6.3.7.

Example 6.3.10. Consider the case where for u, u′ ∈ [c] there is a unique prime closure

A = cl(u, u′) such that if B = cl(u, v) is prime for some v ∈ [c], then A = B. Then if

C = cl(x, x′) is prime for x, x′ ∈ [c], by Theorem 6.3.7, C must also unique in this sense.

119

6.4. EXTENDED MODULAR DECOMPOSITION METHOD

6.4 Extended modular decomposition method

At this point we note that the only currently known graphs for which the recursive k-dim

WL method fails (in that it fails to recursively partition the relevant vertex set to its

orbits) belong to Mk, containing non-isomorphic k-equivalent subgraphs. In particular,

to the knowledge of the author, k-equivalent graphs of the general type in [3] and [4]

are the only graphs for which the recursive 3-dim WL method is known to fail, in that

the only known pairs of non-isomorphic 3-isoregular graphs with the same parameters

are vertex-transitive. With this in mind, in proposing a method of dealing with these

specific counterexample graphs, we will assume for the purposes of this work (and in

particular, the following proposed decomposition method) that prime k-equivalent graphs

are characterised by the recursive (k + 1)-dim WL method (where k ≥ 3). One reason

why this assumption might not be considered particularly onerous for the purposes of this

work is that, as shown in Section 6.1, if the original expander graph used to construct the

counterexample pairs in [3] and [4] can be recursively partitioned down to its orbits by

the k-dim WL method, then the (k+ 1)-dim WL method achieves also this for each of the

counterexample pairs themselves.

6.4.1 Preliminary definitions

Definition 6.4.1. The extended modular decomposition method is defined as a process of

isolating relevant CWS subgraphs of a graph.

In particular, the aim is to isolate then characterise the mutually CWS, k-equivalent

subgraphs, these being the components that provably cannot be partitioned to their orbits

by the k-dim WL method. The title of this Section stems from the analogous definition of

a modular decomposition of a graph. The modules of a graph are subgraphs within which

each element has the same set of neighbours among elements outside the module. Modules

can be proper subsets of other modules, hence the term leads to a recursive decomposition

of a graph, with the set of modules of a graph forming a lattice under inclusion.

Similarly the set of CWS subsets of a graph also forms a lattice with respect to inclusion, as

shown below, and can be thought of as a generalisation of the idea of modules, in this case

relative to the colour classes assigned by the k-dim WL method. In this generalisation,

only elements of the CWS subgraph of the same colour class are required to have identical

neighbour sets outside the subgraph.

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Relative to the standard definition of modules, the modular closure of set of vertices

S ∈ V (G) is defined to be the smallest module R ∈ V (G) that contains S. Here, the

term modular closure will instead be defined relative to this generalisation of modules,

according to the following definition.

Definition 6.4.2. The modular closure, or simply closure, of a set of vertices S ∈ V (G),

denoted cl(S), is the smallest CWS subset of G containing S (i.e. the supremum of S,

relative to CWS modules).

This modular closure is defined relative to the colour classes arising from the k-dim WL

method. There is a simple procedure to calculate the modular closure of a set S, introduced

in Section 6.3. Consider any two elements u, v ∈ S in the same colour class of WLk(G).

Then the elements (d(u)∩e(v))∪(e(u)∩d(v)) must also be in cl(S). Recursively performing

this process until membership in cl(S) is stabilised yields a unique cl(S). Hence cl(S) is

well defined.

Definition 6.4.3. A non-trivial CWS subset is defined as one containing at least two

elements of the same colour class.

Observation 6.4.4. The CWS subsets of a graph form a lattice, under inclusion.

Proof. Consider a graph G, containing two CWS subsets A and B. The modular closure

cl(A∪B) defines a unique supremum (in terms of CWS subsets). Consider the intersection

of the two subsets, C = (A∩B). Then cl(C) ⊆ A, cl(C) ⊆ B, and cl(C) = sup(C), hence

the modular closure cl(A ∩B) defines a unique infimum.

Recall the primality definition given in Section 6.2.1.

Definition 6.2.2. A subgraph S ⊆ G will be termed prime if it has no proper, non-trivial

CWS subgraphs, and is itself a non-trivial CWS subgraphs of G. This is defined implicitly

with respect to the k-dim WL method.

Definition 6.4.5. Given a k-equivalent graph G with colour classes corresponding to

WLk(G), a set of colour classes C = c1, c2, . . . of G will be termed to be trivial in G if

the graph induced on cl([C]) is not k-equivalent.

Definition 6.4.6. Denote a CWS subset S of G to be non-trivial relative to the colour

class c if S contains more than one vertex belonging to [c].

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6.4.2 Graphs under consideration

Recall that the set of graphs Mk is defined in terms of a set of k-equivalent subgraphs

Si ⊂ G that are CWS connected with respect to the colour classes of WLk(Si) |Si , rather

than the colour classes of WLk(Si) |G. Hence it is possible that distinct colour classes of

such an induced graph Si will be merged in G, in that ∃u, v ∈ V (Si) such that

WLk(u) |Si 6= WLk(v) |Si , but

WLk(u) |G= WLk(v) |G .

In order to simplify the analysis of the decomposition method that follows, it will be

defined to act on a subset of Mk in which the properties of WLk(G) are constrained

relative to the colour classes associated with the induced graphs Si.

In particular, it will be defined to act on the set of graphs M′k ⊂Mk defined as follows.

Definition 6.4.7. M′k consists of the graphs G ∈ Mk for which the following further

properties hold.

1. Consider a set of vertices v1, . . . , vr ⊂ V (G) with the properties

WLk(vi) = c,∀i ∈ [r], and

d(vi)\vj = d(vj)\vi∀i, j ∈ [r]. (6.29)

No such set of ‘CWS cliques’ exist in G.

2. Prime CWS subgraphs of G are unique, in the sense that for any x, y, z belonging to

the same colour class ofG, cl(x, y) and cl(x, z) are both prime only if cl(x, y) =

cl(x, z).

The first condition removes the possibility of G containing modules in which all elements

belong to the same colour class, and the graph induced on the module is either complete

or empty. Every subset of such a module is also a CWS subset of G. The first and second

conditions are included to simplify the analysis of the decomposition algorithm presented

later in this section. While they are listed as assumptions, we will see that both can be

enforced without loss of generality, by canonically altering a given graph in Mk.

Before demonstrating that the first and second assumptions can be assumed to hold with-

out loss of generality, further definitions will be required. Consider the following process

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of canonically contracting a CWS subgraph S of a graph G, replacing S by a single vertex,

with the resulting graph labelled by G′.

Definition 6.4.8 (Canonical Contraction). For each colour class ci ∈ WLk(G) with

members in S, denote

[ci]S = v ∈ [ci] : v ∈ S.

Since S is a CWS subgraph of G, a vertex w ∈ V (G\S) can be said to be connected to a

colour class of S, in that if v, w ∈ E(G) for some v ∈ [ci]S , then v′, w ∈ E(G) for all

v′ ∈ [ci]S .

For each w ∈ V (G\S), denote by wS = (cr, γ1), . . . , (ct, γt) the set of colour classes ci

such that w is connected to [ci]S by edges of colour γi. The canonical contraction of G

relative to a subgraph S ⊂ G, resulting in a graph G′S , proceeds as follows.

(i) Replace S by a single vertex x coloured by the isomorphism class of S.

(ii) Replace the edges connecting [ci]S to w ∈ V (G\S) by a single edge x,w coloured

by ci, for all such w.

(iii) Where w ∈ V (G\S) is connected to multiple colour classes in S, replace the resulting

multiple edges by a single edge coloured by each such colour class, such that:

(iv) Edges w, x and w′ , x have the same colour if and only if Sort(wS) = Sort(w′S).

Hence this contraction replaces a CWS subgraph S by a single vertex, while preserving

all information regarding the isomorphism class of S and its connections to G\S, such

that given graphs G and H with subgraphs S1 and S2 canonically contracted respectively,

G′ ∼= H ′ if and only if

S1∼= S2 and G ∼= H.

The contraction is then unbiased (or canonical) in an analogous sense to the unbiased

extension of Section 6.1.

Given a graph G ∈Mk, consider the following two contractions of G.

Construction 6.4.9. Firstly, for each colour class ci ∈ WLk(G), let S = S1, . . . , Stbe the set of all maximum CWS cliques in G for which all subsets of each Si are also

CWS subgraphs of G (in other words, precisely the CWS subgraphs described in (1) of

Definition 6.4.7), where each Si is maximal in the sense that no S′i ⊃ Si exists with the

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same property. Note that the set of such cliques can be efficiently found for any graph G.

Replace all such Si by a single vertex as in Definition 6.4.8, repeat this process recursively

until the result is stabilised, and label the resulting graph by G1.

Definition 6.4.10. Given a graph G, G1 is the resulting graph in which all maximum

CWS cliques are recursively contracted to single vertices as in Construction 6.4.9 above,

such that G1 contains no such subgraphs.

Construction 6.4.11. Secondly, given a subgraph S ⊂ G1, let [ci]S = v ∈ [ci] : v ∈ Sdenote the set of vertices of S ⊂ G1 belonging to colour class ci. Consider a CWS subgraph

R ∈ G1 with the following properties. For all i, j, where v ∈ [ci]R and cj ∈WLk(G1),

d(v) ∩ [ci]R = [ci]R or ∅,

|d(v) ∩ [cj ]R| = 0, 1, |[cj ]R| − 1 or |[cj ]R|. (6.30)

Such a CWS subgraph R has the property that any prime CWS subgraphs of R have

exactly two elements from each colour class in R. Any two prime CWS subgraphs of R

are either vertex disjoint, equal, or have an intersection comprising exactly one element

from each colour class in R. Further, given non-equal prime CWS subgraphs cl(x, y) and

cl(x, z) in R, where x, y, z ∈ [ci]R for some i, the intersection Rx = cl(x, y)∩cl(x, z)is independent of the particular y, z chosen. Finally, the set Rx : x ∈ [ci]R partitions R.

Note that the set of subgraphs R satisfying the above properties can be efficiently found

for any graph G, and this set forms a lattice in G with respect to inclusion.

Replace each such vertex-disjoint Rx, belonging to such a subgraph R ⊂ G1, with a single

vertex as in Definition 6.4.8, repeat this process recursively until the result is stabilised,

and label the resulting graph by G2.

Definition 6.4.12. Given a graph G1, G2 is the resulting graph in which all subgraphs

of the type described above (and those associated with G1) are recursively contracted as

in Construction 6.4.11, such that G2 contains no such subgraphs.

Theorem 6.4.13. Let G be a graph in Mk, and let G1 and G2 be the graphs resulting

from G by the contractions described above. Then the following statements hold.

(i) G1 satisfies condition (1) of Definition 6.4.7.

(ii) G2 satisfies conditions (1) and (2) of Definition 6.4.7.

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Proof. By definition, if G1 contains such CWS cliques, then G1 is not stable, in that the

contraction described can be applied to G1 resulting in some graph (G1)1 6= G1. Hence

(i) holds.

Let A = cl(x, y), B = cl(x, z) be prime CWS subgraphs of G2, where x, y, z ∈ [ci] for

some colour class ci ∈WLk(G2).

Let x, y /∈ cl(A\x, y). Then A\x, y contains at most one vertex of each colour class,

else A is not prime. Similarly, either A = x, y or there exists a v ∈ A, v /∈ x, y such

that v ∈ d(x), v ∈ e(y) or v ∈ e(x), v ∈ d(y). Since A is prime, d(x)∩A = d(y)∩A, however

this contradicts the assumption that x, y ∈ [ci], since x and y must have a different number

of neighbours in the colour class of v. Hence no such v exists, and A = x, y. As no such

CWS cliques exist in G1, this is a contradiction, and so we must have x, y ∈ cl(A\x, y).

Now A∩B contains at most one vertex from each colour class in A and B (else A,B are not

prime). Since x, y, z ∈ [ci], for any colour class cj , |d(x)∩ [cj ]| = |d(y)∩ [cj ]| = |d(z)∩ [cj ]|.In particular, x and y have the same connections outside A, x and z have the same

connections outside B, and y and z have the same connections outside (A ∪B)\(A ∩B).

Denote [cj ]A = [cj ] ∩ A. Then d(x) ∩ [cj ]A = d(y) ∩ [cj ]A for each colour class cj , and

similarly for x, z in B and y, z in (A ∪B)\(A ∩B).

Consider a colour class cj with elements in A∪B, where J = [cj ]A∪B. Let J ′ = J∩(A∩B).

If J ′ 6= ∅, then J ′ = v for some v ∈ (A∩B). Then either J\v ⊂ d(v), or J\v ⊂ e(v),

and hence the graph induced on J is either complete or empty. Trivially, |J ∩A| = |J ∩B|holds.

Consider a second colour class cl, where L = [cl]A∪B, such that cl also has non-empty

intersection, L ∩ (A ∩ B) = v′. As in the connections within [cj ]A∪B, if v′ ∈ d(u) for

some u ∈ [cj ]A, then L ∩ B ⊂ d(u), which in turn implies (L ∩ B)\v′ ⊂ d(u′) for all

u′ ∈ [cj ]A, and v′ ∈ d(u′) for all u′ ∈ ([cj ]A\v. Hence |d(v′)∩ J | ≥ |J | − 1, implying that

|d(w) ∩ J | ≥ |J | − 1 for all w ∈ K.

So far we have established a basic structure for the connections between and within colour

classes of A and B. Namely, if A ∩ B contains a vertex v ∈ [cj ], then the graph induced

on [cj ]A∪B is either complete or empty. Further, if v′ ∈ [cl] such that v′ ∈ (A ∩ B), then

the connections between [cj ] and [cl] within A∪B are either uniform or ‘almost uniform’,

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in that:

∀u ∈ J, |d(u) ∩ L| = |L| or |L− 1|, and hence

∀u ∈ L, |d(u) ∩ J | = |J | or |J − 1|. (6.31)

Given these constraints on A ∪B, consider the iterative process of constructing A and B

from x, y and x, z respectively, described in Section 6.3. At the first step, consider

the set D = v1, . . . , vt, where for all vi ∈ D,

vi ∈ (d(x) ∩ e(y)) or vi ∈ (d(y) ∩ e(x)). (6.32)

Let vi ∈ [cj ] for some vi ∈ D. As the connections between x, y and [cj ]A are uniform or

almost uniform in the above sense, either d(x) ∩ [cj ] = d(y) ∩ [cj ] or |D ∩ [cj ]| = 2, one

vertex of which necessarily belongs to A∩B. Iterating this process, if |[cm]∩ (A∪B)| > 0

for any colour class cm, then

|[cm] ∩ (A ∪B)| = 3,

|[cm] ∩ (A ∩B)| = 1, and (6.33)

|[cm] ∩A| = |[cm] ∩B| = 1.

Hence A∩B is precisely the (trivial) CWS subgraph type that is contracted in the process

described in 6.4.12 above. As no such subgraphs exist in G2, no such intersecting prime

CWS subgraphs cl(x, y) and cl(x, z) exist, and hence prime CWS subgraphs of G2

are in this sense unique.

Corollary 6.4.14. Any G ∈ Mk can be assumed without loss of generality to satisfy

conditions (1) and (2) of Definition 6.4.7, by instead considering the graph G2 obtained

by recursively applying the two contraction process to G described above.

Note that the graphs of interest within M′k contain those known to not be partitioned

down to their orbits by the k-dim WL method. For such a graph G, there will exist at

least one colour class c ∈WLk(G) consisting of two or more orbits of Aut(G).

Lemma 6.4.15. Let G ∈ M ′k contain two vertex-disjoint, non-trivial CWS, mutually k-

similar subgraphs S1, S2, such that u ∼k v within G, for some u ∈ V (S1), v ∈ V (S2).

Then cl(u, v) is not prime.

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Proof. Assume that cl(u, v) is prime, and let A = cl(u, v)∩S1 and B = cl(u, v)∩S2.

A and B are each CWS subgraphs of G, so either cl(u, v) is not prime, or A and B each

contain at most one vertex from any given colour class.

Let u, v ∈ [c] for some colour class c ∈ WLk(G). Then for all x ∈ A, x ∈ [ci] there exists

a y ∈ B such that y ∈ [ci] and vice versa. Hence A and B are mutually k-equivalent. Let

R1(i) and R2(i) represent the sets of vertices of colour class ci in S1 and S2 respectively.

Since S1 and S2 are each CWS subsets of G,

d(u) ∩R2(i) = R2(i) or ∅, (6.34)

and similarly for d(v). As |cl(u, v) ∩R1(i)| = 1 or 0, then

d(u) ∩R1(1) = R1(1)\u or ∅, and

|d(u) ∩R1(i)| = |R1(i)|, 0, 1 or |R1(i)| − 1, (6.35)

and similarly for d(v) in S2. Note that since S1 and S2 are mutually k-equivalent,

|d(u) ∩R1(i)| = |d(v) ∩R2(i)| ∀i. (6.36)

Hence if S1 contains another vertex w 6= x of colour class [c], cl(x,w) is also prime,

and the sets A and B are precisely those contracted to a single vertex by the contraction

process of Definition 6.4.12. Hence no such prime closure cl(x, y) exists.

6.4.3 Decomposition method

Following the above introductory definitions and properties, we can now define the fol-

lowing method of decomposing G ∈M ′k into the mutually CWS, mutually k-equivalent Si

subsets that are prime.

Algorithm 6.4.16 (Decomposition). Given a graph G ∈ M′k, we define the following

extended modular decomposition method:

1. Act on G with the k-dim WL method, determining the colour classes WLk(G) (rel-

ative to which the CWS subsets are defined).

2. Choose a vertex u (without loss of generality) from the lexicographically smallest

colour class, c.

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3. For each vertex v ∈ [c], v 6= u, calculate cl(u, v), and determine if this closure is

prime.

4. If no such prime closure exists, remove this colour class from consideration, returning

to step (3). Else record the unique prime closure cl(u, v) (for some appropriate

v).

5. Repeat steps (2)-(4) recursively on the remaining elements of [c] (not currently con-

tained in a prime closure) until all elements of [c] are associated with a prime CWS

subgraph.

6. Repeat steps (2)-(5) recursively for the remaining colour classes not incorporated in

the previous prime closures.

Lemma 6.4.17. Let G ∈ M ′k be a graph as in definition 6.4.16 above. For each colour

class c ∈WLk(G), the following hold:

(i) There exists a partitioning V1, V2, . . . of [c] such that all cl(Vi) are prime, and all

cl(Vi) subgraphs are mutually k-similar.

(ii) If a cl(Vi) as above contains vertices of colour class c′, then cl(V1), cl(V2), . . . par-

titions [c′] in the same manner.

Proof. Let u, v ∈ [c] such that cl(u, v) is prime. Then for all w ∈ [c], cl(u,w) is prime

if and only if w ∈ cl(u, v). Similarly, by Theorem 6.3.7, each vertex u ∈ [c] has sets of

pairwise CWS closures sharing several properties. In particular, for all u ∈ [c] there is a

unique prime closure containing u, and furthermore for any u, v ∈ [c], the prime closures

containing u and v respectively are either m-equivalent (for some m ≥ k) or isomorphic.

Hence (i) holds, and as a corollary, (ii) holds.

Corollary 6.4.18. Any two prime CWS subgraphs resulting from Algorithm 6.4.16 which

have overlapping colour classes are k-similar.

The set of colour class partitions outputted by the above decomposition method are used

to characterise the graph G ∈M ′k, according to the following ‘wrapper’ algorithm.

Algorithm 6.4.19 (Reduction). Denote the set of prime, vertex-disjoint, CWS sub-

graphs obtained from the method of Algorithm 6.4.16 by TG.

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1. Apply the recursive k-dim WL method to each T ∈ TG, obtaining a graph certificate

characterising each†.

2. Canonically contract each T ⊂ G as described in Definition 6.4.8, replacing it with

a single vertex vT coloured by the isomorphism class of T , and similarly contracting

the edges incident with T as in Definition 6.4.8.

3. Label the resulting graph G(1) (where G := G(0)).

4. Recursively repeat steps (1)-(3), obtaining the graph G(i) after the ith repetition,

until the resulting graph is stabilised, such that G(t) = G(t+1).

Theorem 6.4.20. Applying the process of Algorithms 6.4.16 and 6.4.19 to graphs G and

H in M′k,

G(i) ∼= H(i) if and only if G ∼= H.

Proof. As the prime CWS subgraphs of a graph G are unique (in the sense of Definition

6.4.7), G ∼= H only if TG = TH . Hence, as the contraction process of Definition 6.4.8

preserves isomorphism, G(1) ∼= H(1) if and only if G ∼= H. Similarly G(i) ∼= H(i) if and

only if G(i−1) ∼= H(i−1), and the result follows.

Theorem 6.4.21. If G(t) = G(t+1) then G(t) is characterised by the recursive k-dim WL

method.

Proof. Let G(t) contain some CWS k-equivalent subgraph A. If A is not prime, then there

exists some prime CWS subgraph S ⊂ A such that S ∈ TG(t) . However this implies that

G(t+1) 6= G(t), hence no such S exists, and A is prime.

Hence if any subgraph A ⊆ G(t) is m-equivalent for some m ≥ k, then A is prime.

By assumption, prime k-equivalent graphs are characterised by the recursive k-dim WL

method, and so the result follows.

Definition 6.4.22. The smallest t such that G(t) = G(t+1) is termed the recursion depth

of G.

Lemma 6.4.23. For any G ∈M ′k, the following hold:

(i) G has recursion depth bounded above by O(log |V (G)|).†By assumption, prime k-equivalent graphs are characterised by the recursive k-dim WL method. Fur-

ther, we assume that non-k-equivalent graphs are also characterised by the recursive k-dim WL method.

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(ii) G(i) can be calculated from G(i−1) in time O(poly(|V (G(i))|)).

Proof. For (i), note that the elements of TG partition the colour classes of interest. More-

over, elements of TG(1) must contain vertices corresponding to the elements of TG, and two

such vertices v1, v2 are in the same colour class if and only if the corresponding T1, T2 ∈ TGare isomorphic and mutually CWS. Now each T ∈ TG contains at least 3 vertices, so simi-

larly each T ∈ TG(1) contains at least two vertices corresponding to mutually CWS T ∈ TG,

and hence corresponds to at least 7 vertices of G. Hence each T ∈ TG(i) corresponds to at

least 2i+2 − 1 vertices of G, and (i) follows.

To show that (ii) holds, note briefly that each step in Algorithm 6.4.16 can be done in

time O(poly(|V (G(i))|)). Namely, Algorithm 6.4.16 consists of first applying the k-dim

WL method, requiring time O(nk+1) for a graph on n vertices, followed by calculating the

modular closure of at most O(n2) pairs of vertices (and checking each closure for primality),

which can also be accomplished in polynomial time. The implicit extra step of contracting

requisite subgraphs of G ∈ Mk to form a graph in M ′k, as detailed in Constructions 6.4.9

and 6.4.11 is also trivially accomplished in time O(poly(n)), for a graph of size n, hence

(ii) follows.

Corollary 6.4.24. Hence the combination of Algorithms 6.4.16 and 6.4.19 characterises

the graphs of Mk that satisfy the assumptions recalled in 6.4.4 in polynomial time.

The significance of these results, and of the assumptions made, will be discussed in the

following sections.

6.4.4 Discussion of Assumptions

The decomposition method comprising Algorithms 6.4.16 and 6.4.19 relies on assumptions

regarding the properties of the input graphs, relative to the k-dim WL method. Specifi-

cally, any graph G ∈M ′k will be characterised by Algorithm 6.4.19 providing the following

assumptions hold:

1. Non-k-equivalent graphs are characterised by the recursive k-dim WL method (where

k ≥ 3).

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2. Prime k-equivalent graphs can be characterised by the recursive (k + 1)-dim WL

method.

Non-k-equivalent graphs are precisely those which are assigned unique certificates (and

hence characterised) by the k-dim WL method. This does not necessarily imply that the

recursive k-dim WL method will characterise such graphs, in that they may not be refined

to their orbits at each step. In particular, 3-isoregular graphs that aren’t vertex-transitive,

or 4-isoregular graphs that aren’t 2-transitive will not be refined to their orbits by the 3-

dim and 4-dim WL method respectively. Whether this assumption holds in general is

unknown, however note that as shown in Section 6.1, the known k-equivalent graphs are

characterised by the recursive (k+ 1)-dim WL method, and hence no counterexamples to

assumption (1) are yet known.

Similarly the only known prime k-equivalent graphs are those described in [3] and [4],

and so likewise no counterexamples to assumption (2) are known. One possible method

of constructing prime k-equivalent graphs that do not satisfy this assumption might be

the following: Extend a k-equivalent graph as described in Section 6.1, using unbiased

gadgets with vertices corresponding to 3-tuples of the original graph. In this case the the

success or failure of the (k+1)-dim WL method is unclear, although the results of Section

6.1 imply that the (k+ 2)-dim WL method will suffice to refine the resulting graph to its

orbits.

6.5 Conclusions and Future Work

The goal of this work is to explore the question,

“Has the k-dim WL method been proven to not solve GI?”

This has indeed been proven regarding a direct implementation of the k-dim WL method

[3, 4]. However one of the essential characteristics of these proofs are that the counterex-

ample graphs found (termed k-equivalent graphs) possess very specific properties. As a

result, this work explores the possibility of exploiting these restrictive properties to design

an extension to the WL method that can characterise these graphs.

We show that the k-equivalent graphs constructed in [3, 4] are individually characterised

by the recursive (k + 1)-dim WL method. The results of [3, 4] are then expanded upon,

constructing a family of k-equivalent graphs not refined to their orbits by the recursive

131

6.5. CONCLUSIONS AND FUTURE WORK

k-dim WL method. Given this family of graphs, we establish various related properties

of the k-dim WL method, and construct an algorithm that canonically decomposes such

k-equivalent graphs into graphs for which the recursive k-dim WL method succeeds, hence

characterising the original graph. In the process an extension to the k-dim WL method

is constructed which efficiently characterises all known k-equivalent graphs, and hence

represents a new approach to addressing the GI problem

The known k-equivalent graphs were introduced in [3] and [4] as token counterexamples

to the k-dim WL method. Minor variations to each family of graphs can be trivially

constructed while preserving the property of k-equivalence. To the extend that this work

establishes an extension to the WL method characterising these graphs, it removes the

known, token counterexamples. However its significance lies in the fact that it does not

simply address these counterexamples in isolation, but holds for all k-equivalent graphs

for which the assumptions of Section 6.4.4 are satisfied.

Constructing counterexamples to this extension would require finding graphs with novel

properties, for which these assumptions do not hold. In particular, where k ≥ 3 and

m ≥ k, counterexamples must belong to at least one of the following categories:

1. Prime m-equivalent graphs that cannot be refined to their orbits by the k-dim WL

method.

2. Alternatively, prime m-equivalent graphs that cannot be characterised by the recur-

sive k-dim WL method.

3. Non-k-equivalent graphs that cannot be refined to their orbits by the k-dim WL

method

No such graphs have yet been found, hence a proof that this extended method does not

solve GI would require finding graphs with novel properties. In particular, while there is

no reason to suspect that these assumptions do hold for general graphs, they have been

shown here to hold for all known counterexamples to the recursive k-dim WL method.

In relating these results to general k-equivalent graphs, we note that not much is known re-

garding possible general properties of such graphs, as the known cases were, as mentioned,

introduced as token counterexamples, and with the exception of the work of [16, 123] the

properties of k-equivalent graphs have not been explored further.

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6.5.1 Open Questions

As mentioned above, very little is known regarding possible general properties of k-

equivalent graphs. Trivial k-equivalent graphs do exist, namely graphs which are k-

isoregular (also known as k-tuple regular), however such graphs have been completely

characterised for k ≥ 5. One interesting open question relating to such graphs is whether

t-isoregular graphs exist for t ∈ 3, 4 that cannot be characterised by the recursive 3-dim

WL method. To find such a graph it would suffice to find either a non-vertex-transitive

3-isoregular graph or a 4-isoregular graph that is not distance-transitive, however to the

knowledge of the author no such graphs are known to exist.

Apart from the consideration of k-isoregular graphs, what other kinds of potentially k-

equivalent graphs exist? In particular, are there any general properties that such graphs

must possess (in addition to k-equivalence) that restrict possible types? The two known

families of k-equivalent graphs share several important properties, and in essence ‘obtain’

their k-equivalence in the same way, via the expansion of graphs with large separator sizes.

However these properties also make the graphs amenable to classification by the extended

WL scheme constructed here.

One question relating to general properties of k-equivalent graphs was previously raised

in [16], in which the possibility that the k-dim WL method (for some bounded k) may

suffice to distinguish pairs of strongly regular graphs was raised. It was noted that strongly

regular graphs have particularly simple cellular closures (related to the WL method in

Section 5.4). In particular, the coherent configuration corresponding to a strongly regular

graph with adjacency matrix A has only three basis relations, I, A, (J − I−A), where I

is the identity matrix and J the all-1 matrix. An interesting open problem is an analysis

of the k-extended cellular closures of strongly regular graphs, relating to the question of

whether k-equivalent graphs can be strongly regular (for some bounded k > 2).

Related to the family of k-equivalent graphs are graphs which the k-dim WL method fails

to refine to their orbits. General properties of such graphs are also unknown; trivial cases

can be formed via constructing a graph containing mutually CWS k-equivalent subgraphs,

such as those belonging to M ′k, however do other graphs exist for which the recursive k-dim

WL method fails?

Regarding the distinction between refining a graph to its orbits via the k-dim WL method,

and determining the automorphism group by recursively stabilisation using the recursive

k-dim WL method, the following question occurs. If the k-dim WL method (for k ≥ 3)

133


refines a graph to its orbits, must the recursive k-dim WL method characterise the graph.

In other words, will the vertex-stabilised graph also be refined to its orbits by the k-dim

WL method?

Finally, for which graphs do the assumptions of Section 6.4.4 not hold?

134

Part III

Adinkras and Off-Shell

Representations of

Supersymmetry

135

Introduction

Based on B. L. Douglas, S. J. Gates, Jr. and J. B. Wang’s

paper, “Automorphism Properties of Adinkras” [129], available

at arxiv:1009.1449 [hep-th], currently under submission.

While studying the general graph isomorphism techniques discussed in Part II, a problem

of graph characterisation in the research area of supersymmetry came to the attention of

the author. In particular, beginning with the work of Faux and Gates [5], a graphical

technique for representing off-shell representations of supersymmetry by members of a

family of graphs called Adinkras has been developed. Combining elements from a variety

of disciplines, this work has been progressing rapidly for several years, forming interesting

connections and collaborations between typically separate areas.

One of the central problems in this research area is the classification of Adinkras, graphs

which possess topology of a binary character. Crucially, as with the graphs considered in

Chapter 3, they also possess a high degree of symmetry, enabling them to be characterised

by a number of parameters growing at most logarithmically with the number of vertices.

What started as an attempt to apply a variation on the techniques of Part II to characterise

this family of graphs has progressed to a study involving previously unfamiliar fields, such

as supersymmetry, representation theory and coding theory, and has led to the continuing

pursuit of a variety of questions.

In a way, this completed the often circuitous journey undertaken in this work, starting

with techniques in theoretical physics in Part I, specifically relating to quantum walks,

applying these to a purely graph theoretic problem in Part II, and ending by in turn

applying graph theoretic techniques to a theoretical physics problem, as will be described

in this Part. However, this mapping from physics to maths and back again did not square

136

to the identity, and one unintended consequence is that we find ourselves in a completely

different area of physics!

Hence some introductory material will again be required. Rather than devote large sec-

tions of this thesis to a comprehensive introduction to supersymmetry, and in particular

supersymmetric representation theory, a brief introduction will instead be provided, for

the purposes of a familiarisation with the context in which Adinkras arise. References to

comprehensive reviews of these subjects will be provided, however for our purposes little

background knowledge of representation theory will be required to frame the discussion on

Adinkras, which will instead predominately take place within the context of graph theory.

The concept of Adinkras has its roots in the work of [28,29,130,131], beginning as a series

of observations of universal matrix algebra structures that appear to occur in all off-shell

supersymmetrical theories. Subsequently, Adinkras emerged [5] as a graphical means of

representing these matrix algebras. Several recent studies [26, 27, 30, 32, 132, 133] have

continued this work, developing Adinkras as a novel technique with which to approach

the off-shell problem of supersymmetry, essentially encoding representations of supersym-

metry into a particular family of graphs. This graphical encoding has the advantage of

allowing convenient manipulation of these objects, with the goal of achieving a deeper

understanding of the underlying representations.

Regarding the problem of classifying the family of Adinkra graphs, recent studies [25,

26, 30, 31] have considered various aspects of such a classification, and have succeeded

in ascertaining many of the properties of Adinkras. The works of [26] and [27] relate

topological properties of Adinkras to doubly even codes and Clifford algebras, and are

discussed in detail in the following Chapter. The work of [31] relates topological properties

of Adinkras to Betti polynomials. As the ultimate goal of this area of work is to provide

a definitive classification of all off-shell supersymmetric theories, the appearance of new

unexpected discoveries is encouraging, suggesting that this goal might be within reach,

despite a degree of pessimism surrounding this problem that has remained unsolved for

over 30 years.

The main focus of Part III is the characterisation of the automorphism group of general

Adinkras. I will relate to each of two possible definitions of isomorphism a classification

of the automorphism group of Adinkras in terms of their associated doubly even codes.

Conversely, the code in question will also be characterised in terms of local properties of

the Adinkras, hence serving to classify equivalence and isomorphism classes of Adinkras,

137

together with their automorphism group, in terms of a set of efficiently computable local

parameters.

Using the automorphism group techniques developed, previous results from the work of [32]

proposing a characterisation of Adinkra degeneracy via matrix products will be extended

and generalised. In particular, we identify all cases in which Adinkras can exhibit this form

of degeneracy, and construct a proof that a generalised version of these matrix products

characterises this degeneracy for all Adinkra dimensions N .

To demonstrate the effectiveness of the method of automorphism group classification con-

structed, we explicitly construct an algorithm using these results to partition Adinkras

into their equivalence and isomorphism classes.

The structure of this Part is as follows. Firstly a brief overview to supersymmetric rep-

resentation theory is provided in Chapter 7, detailing the history and significance of the

‘off-shell problem’ mentioned above, followed by a formal definition of Garden algebras

(the matrix algebras alluded to above) and Adinkras. Previously established properties of

Adinkras are derived and discussed in the remainder of this Chapter.

After this introductory Chapter, the original work of this Part comprises Chapter 8, which

begins by establishing the form of a general Adinkra’s automorphism group relative to its

corresponding doubly even code in Section 8.1. Section 8.2 deals with the characterisation

of degeneracy via a matrix polynomial, as described above, generalising the results of [32].

Section 8.3 employs the results of Section 8.1 to partition Adinkras into their isomorphism

classes, as well as ascertaining the requisite alterations to the automorphism group results

of Section 8.1 if an alternative definition of isomorphism is considered. Finally, in Section

8.4 numerical methods employed in the preceding sections are detailed.

138

Chapter 7

SUSY and Adinkras

7.1 A Brief Introduction to Supersymmetry

The study of theoretical physics is intimately connected to the concept of symmetries;

transformations that leave a system unchanged in some important respect. Physical laws

such as conservation of energy and momentum arise from particular symmetries of space-

time. For instance, invariance of physical systems under translations in time implies

conservation of energy. The Standard Model of particle physics incorporates such known

symmetries, providing a theory with remarkable predictive power, consistent with a va-

riety of experimental results [134]. The Standard Model has several limitations however,

among them a failure to consistently incorporate gravity and an inelegance relating to the

‘hierarchy problem’ [135–137]. Several such limitations can be addressed via extensions to

the model which incorporate supersymmetry.

The concept of supersymmetry has been around in the physics literature for more than

30 years. Extensive reviews on the phenomenology and technical aspects of supersym-

metry can be found in [138, 139] and [140, 141] respectively. Briefly, a supersymmetry

transformation maps bosons to fermions, and vice versa. The operator performing this

transformation is denoted by Q, termed a supercharge, where

Q|boson〉 = |fermion〉, Q|fermion〉 = |boson〉.

If supersymmetry exists, each fundamental particle of matter must have a related super-

partner particle. An unbroken supersymmetry would require such superpartners to have

identical mass. However no such particles are observed, so if supersymmetry exists it

139

7.1. A BRIEF INTRODUCTION TO SUPERSYMMETRY

must be spontaneously broken, allowing higher mass superpartners [142]. Together with

providing a useful general framework for high energy theories, supersymmetry is also a

required component of all known physically realisable string theories [143].

We consider the following general framework for supersymmetry (SUSY). Consider a

Hamiltonian H acting on a Hilbert space H. A one-dimensional N -extended supersymme-

try comprises N distinct supercharges (alternatively termed supersymmetry generators),

Q1, . . . , QN , and involves a single, time-like coordinate, labelled by τ . The corresponding

supersymmetry algebra (or superalgebra), without a central charge, satisfies the relations

QI , QJ = 2δIJH, [H,QI ] = 0, (7.1)

where the Hamiltonian H can be identified with i~∂τ , and δIJ is the Kronecker delta. A

superalgebra is simply a standard algebra combined with an additional Z2 grading. Hence

together with the self-adjoint H and QI operators, a further self-adjoint operator Γ is

included (termed a grading operator or fermion number operator), such that

Γ2 = +1, Γ, QI = 0. (7.2)

The +1 and −1 eigenspaces of Γ are associated with bosonic and fermionic fields respec-

tively, hence the term fermion number operator. Hence if the fields are grouped into bosons

and fermions, Γ takes the form

Γ =

1 0

0 −1

. (7.3)

Similarly, relative to this representation the QI are off-diagonal, mapping bosons to

fermions and vice versa, such that each QI can be represented by

QI =

0 LI

RI 0

, (7.4)

relative to operators LI and RI , mapping bosons to fermions and fermions to bosons

respectively.

A representation of a supersymmetry algebra is termed a supermultiplet. The components

of a supermultiplet are split into bosonic and fermionic fields, related by the supersym-

metry generators QI . There is a long-standing problem in the representation theory of

superalgebras known as the ‘auxiliary field problem’, which is essentially the problem of

140

CHAPTER 7. SUSY AND ADINKRAS

finding a representation of a given superalgebra that is independent of the associated su-

persymmetric field theory. Such a representation is said to be formulated ‘off-shell’, hence

this problem is alternatively known as the ‘off-shell SUSY problem’.

An off-shell representation of a superalgebra is one in which the basis fields represent the

superalebgra without additional dynamical constraints. Unlike Yang-Mills theories [144,

Ch. 9], in which off-shell representations of the Lie algebras have been classified [25], very

few supersymmetric field theory admit known off-shell solutions. The cases where off-shell

representations have been found are almost uniformly of low space-time dimension, less

than or equal to four [25].

For almost all theories involving greater than four space-time dimensions, the correspond-

ing superalgebras have known representations only when the component fields are con-

strained to satisfy differential equations corresponding to laws of motion - dynamical

constraints such that the fields satisfy Euler-Lagrange equations. Such representations

are termed on-shell.

The off-shell problem has remained unsolved for over 30 years, however recently new

techniques with which to approach it have been developed. In particular, beginning with

the work of [28, 29, 130, 131], a subset of the real Clifford algebras termed the ‘GR(d,N)’

algebras (or ‘garden algebras’ ) was related to generalised scalar and spinor supermultiplets.

These algebras seem to appear in all off-shell supersymmetric theories, and are conjectured

to provide the basic components necessary for a rigorous theory of space-time SUSY

representations [29]. Complementary to this work, a family of graphs termed Adinkras

emerged as a graph-theoretic tool for studying these garden algebras [145]. The approach

has since been further developed in the works of [25–27, 30], and references therein. The

ultimate goal in developing these tools is to provide a definitive classification of all off-

shell supersymmetric theories, and in this respect Adinkras provide a convenient method

of encoding and manipulating these objects.

7.1.1 Garden Algebras and Adinkras

Garden algebras attempt to embed supersymmetry representations within the structure

of Clifford modules [146]. The idea is based on the following two assumptions:

1. The essential properties of general SUSY field theories are captured by the cor-

responding one-dimensional theories related by dimensional reduction (for details,

141


see [28,29,130,131,145,147]).

2. A complete representation theory of one-dimensional superalgebras is encoded in the

representation theory of garden algebras.

The N -extended superalgebra described by Eq. (7.1) can be rewritten in terms of a

parameter-dependent transformation associated with the supercharges. Specifically, we

define

δQ(ε) := −i εIQI , (7.5)

where εI is a set of N real anticommuting parameters (here Einstein summation notation

is used). In terms of this operator, Eq. (7.1) can be rewritten equivalently as

[δQ(ε1), δQ(ε2)] = 2 i εI1 εI2∂τ . (7.6)

Note that here the parameter superscripts indicate supersymmetries (where I ∈ [N ] for

an N -extended supersymmetric model), whereas the subscripts simply denote different

choices of the parameter.

Formally, the GR(d,N) algebras, or garden algebras, are defined to be the subset of the

real Clifford algebras with N linearly independent generators γI , I ∈ 1, . . . , N, such

that firstly,

γI , γJ = 2δIJ1, ∀ I, J ∈ [N ], (7.7)

and secondly an additional generator F exists, with the property

F 2 = 1, F , γI = 0, ∀ I ∈ [N ]. (7.8)

Hence (as demonstrated in [28]) it follows that there exist projection operators

P± =1

2(1± F ) (7.9)

associated with bosonic and fermionic states, such that for all I,

P+γIP+ = P−γIP− = 0, (7.10)

142


and there exist non-zero LI , RI such that

LI = P+γIP−, and RI = P−γIP+. (7.11)

In other words, as in Eq. (7.4), the LI and RI are simply the components of the generators

γI (corresponding to the supercharges of Eq. (7.1)) that relate to boson to fermion and

fermion to boson transformations respectively.

Given the nature of the discussion that follows, focussing on the graph-theoretic inter-

pretation of these results (in terms of Adinkras), it will be convenient for our purposes

to paraphrase the definition of the GR(d,N) algebras (as presented in [5,28,29]), making

the following additional assumptions (without loss of generality) regarding the ordering of

component elements.

Consider a Clifford algebra with N generators defined on a vector space V split into two

components, V = VL ⊕ VR (analogous to the splitting of boson and fermion fields in Eq.

(7.3)), such that each generator γI satisfies the matrix representation

γI =

0 LI

RI 0

. (7.12)

In other words, the elements of the representation are ordered relative to the splitting

described above. Then by Eq. (7.7) the set of LI (resp. RI) are simply linearly independent

permutation matrices. Continuing the boson/fermion analogy, the generator F described

in Eq. (7.8) is precisely the fermion number operator, or grading operator Γ described in

Eq. (7.3).

Given this particular restriction of ordering into two components, the garden algebra

definition can be rephrased in the following simplified form, independently to the existence

of F (although also implied by the definition of the generator F [28, 29])

RTI = LI . (7.13)

These generators γI , I = 1, . . . , N then suffice to provide a representation of GR(d,N).

Comparing this to the definition of GR(d,N) encompassed by Eqs. (4.3) and (4.5) of [5],

we see that the restrictions of Eq. (4.3) of [5] result directly from Eq. (7.7) above.

143


By associating the components VL and VR with the space of bosons and fermions respec-

tively, and identifying the generators γI with supercharges, supersymmetry transforma-

tions can be defined on garden algebras. Let φi(τ) ∈ VL and ψj(τ) ∈ VR. Then one such

supersymmetry transformation is given by the relations

δQ(ε)φi = i εI(LI)ij ψj ,

δQ(ε)ψi = εI(RI)ij ∂τ φj . (7.14)

These relations satisfy Eq. (7.6) since LI and RI satisfy Eq. (7.13), and hence they describe

a supersymmetry transformation. Specifically, the relations of Eq. (7.14) describe what is

termed a scalar supermultiplet. Similarly, a spinor supermultiplet can be represented by

the relations

δQ(ε)φi = i εI(LI)ij ∂τ ψj ,

δQ(ε)ψi = εI(RI)ij φj . (7.15)

Note that Eqs. (7.14) and (7.15) differ only in the placement of the derivative ∂τ .

In the case where i, j = 1 (in other words when there exists only a single boson and

fermion), Eqs. (7.14) and (7.15) are the two possible irreducible representations of the one-

dimensional superalgebra. Note that any representation of an N -extended superalgebra

can be decomposed as a set of irreducible one-dimensional superalgebras. Since there

exist only the two such elemental one-dimensional superalgebras, differing only in the

placement of the derivative ∂τ , and any higher-N representations can be viewed as an

ensemble of such elemental cases, a graphical depiction of these representations can be

developed, as explored in the work of [5]. This graphical method exploits the relationship

between Eqs. (7.14) and (7.15), representing n-extended superalgebras by a family of

graphs termed Adinkras, in which each of the irreducible one-dimensional superalgebras of

the decomposition described above are represented by an edge of the graph connecting a

boson to a fermion, oriented according to which of the two one-dimensional superalgebras

applies.

To introduce this method of graphical depiction, we first consider the simplest, N = 1

case. Out of the two possible irreducible N = 1 representations, the irreducible scalar

144


supermultiplet described can be represented by

δQφ = i ε ψ,

δQψ = ε∂τφ, (7.16)

and the irreducible spinor supermultiplet can be represented by

δQ φ = i ε ∂τ φ,

δQ ψ = ε φ. (7.17)

In this case there is only a single boson and fermion, labelled by φ and ψ respectively, and

a single supercharge, hence the subscripts and superscripts of Eqs. (7.14) and (7.15) can

be dropped.

To these supermultiplets we relate the 2-vertex graphs below, being the simplest possible

Adinkras.

(7.16) (7.17)

In each graph, the white nodes represent bosons, and the black nodes fermions. The

edges represent a supersymmetry transformation (between bosons and fermions), such

that an edge (f, g) between a two generic component fields f and g corresponds to the

transformation rule [5]

δQ(ε) f = ±ib ε ∂λτ g, (7.18)

where b = 1 if f is a boson and b = 0 otherwise. Note that Eq. (7.18) is a convenient way of

capturing the relationship between Eqs. (7.16) and (7.17), and encodes both situations. As

suggested by the form of the two graphs above, the relative height of two nodes connected

by an edge is important. In particular, the appearance of the partial derivative in Eq.

(7.18) is dictated by the relative height of f and g, such that λ = 1 if f is higher than g

and λ = 0 if f is lower than g. The factor of ±1 represents a global freedom in the sign of

the relations of Eqs. (7.14) and (7.15); each could be consistently defined with a factor of

145


−1 added to the right hand side of the two relevant relations. This choice is encoded by

the ‘parity’ of edges in an Adinkra, represented graphically by dashed or solid edges.

Before explicitly defining the properties of Adinkra graphs and the related supermultiplets,

we present a further example: the N = 2 case, involving two supercharges. Consider an

N = 2 superalgebra consisting of four component fields, two bosons and two fermions,

related to the Adinkra:

φ1 φ2

ψ2

ψ1

Again the black and white nodes represent fermions and bosons respectively. The distinct

supersymmetries are represented by the colour of the edges, where the green and yellow

edges are parametrised by ε1 and ε2 respectively.

Then applying the transformation rule of Eq. (7.18), the Adinkra encodes the following

set of relations

δQ(ε)ψ1 = εI φI ,

δQ(ε)φI = i εI ∂τ ψ1 + i εJI εJ ψ2, (7.19)

δQ(ε)ψ2 = εIJ εI ∂τ φJ ,

where I, J ∈ 1, 2, I 6= J , and ε represents the Levi-Civita symbol, such that ε12 =

−ε21 = 1. Note that consistency with respect to Eq. (7.6) requires that not all edges can

be solid (or dashed). In particular, since QI , QJ = 0 when I 6= J , there must be a sign

difference between the operations Q1Q2 and Q2Q1, and hence each such length-4 cycle

(in this case there is only one) must contain 1 modulo 4 dashed and solid edges. This

is equivalent to requiring that the product of all signs in the transformation rules of Eq.

(7.19) (along a given direction around the graph) must be −1.

In this way, the N = 1 Adinkras can be combined to form higher N Adinkras, containing

N distinct edge-colours, and hence representing N -extended superalgebras with arbitrary

N . In this work, we deal only with a certain subclass of Adinkras, termed equivalently

‘engineerable’ Adinkras or Adinkras related to ‘Adinkraic’ supermultiplets. Each node

146


of an Adinkra is assigned an integer height corresponding to its engineering dimension

(pg. 446 of [144]), such that connected nodes are always at adjacent heights. Further, the

‘upper’ endpoint of an edge, in the sense of λ in Eq. (7.18), always has a height assignment

greater than the lower endpoint of the edge. An Adinkra is termed engineerable if an

integer height can be assigned to each node, satisfying the above conditions. Adinkras for

which this can not be done in a consistent way are termed Escheric Adinkras [5, 25].

Note that height assignments only encode the relative engineering dimension of component

fields of the related supermultiplet. For example, a given Adinkra can equivalently describe

some supermultiplet related by the transformation rules of Eq. (7.18), or alternatively

the supermultiplet obtained by applying some superderivative operator to this superfield.

For a more detailed discussion of the relationship between Adinkras and supermultiplets,

see [5, 25].

7.2 Formal Definition of Adinkras

7.2.1 Graph-Theoretic Notation

Most of the graph-theoretic notations to be used in the following sections have been

introduced previously; those which have not are defined below. The notation used here

to describe a graph differs slightly from Part II. Rather than denoting a graph by G with

vertex and edge sets V (G) and E(G) respectively, the more standard notation G(V,E)

is used, and when discussed in this context, the vertex and edge sets are simply denoted

by V and E respectively. It is assumed that all graphs referred to are both simple and

undirected.

Definition 7.2.1. A graph G(V,E) is edge-N-partite if its edge set E can be partitioned

into N disjoint sets, such that every vertex v ∈ V is incident with exactly one edge from

each of these N sets.

In other words, a graph is edge-N -partite if it has a 1-factorisation.

As in the preceding sections, a colouring of the edge or vertex set of a graph is a partitioning

of these sets into different colour classes. Formally, this restricts the automorphism group

of the graph to the subset that setwise stabilises these colour classes - the subset that does

not map vertices (resp. edges) in one colour class to vertices (resp. edges) in another.

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7.2. FORMAL DEFINITION OF ADINKRAS

7.2.2 Definition and Properties

Following on from the preceding description of Adinkras, a more complete graph-theoretic

definition will be presented. An Adinkra graph G(V,E) related to a supermultiplet M is

a bipartite, vertex- and edge-coloured graph with the following properties.

Each vertex in V represents a component field of M, and these vertices are coloured in

two ways:

• A colouring corresponding naturally to the bipartition (labelled as bosons and

fermions).

• Each vertex is given a height assignment, hgt: V → Z such that adjacent vertices

are at adjacent heights.

The edges are also coloured in two ways:

• A partition into N colour classes corresponding to an edge-N -partition of the graph.

• An edge parity assignment π : E → Z2. We term these two edge types to be dashed

and solid.

Finally, the connections in the graph are essentially binary in the following manner. Every

path of length two having edge colours (i, j) defines a unique 4-length cycle with edge

colours (i, j, i, j). All such 4-length cycles have an odd number of dashed edges.

We refer to the valence of an Adinkra as its dimension. Hence an Adinkra with N edge

colours (not counting edge parity) is an N -dimensional Adinkra. There will generally be

an assumed ordering of the edge colours from 1 to N , with the term ith edges or ith edge

dimension referring to all edges of the ith colour. The edge parity is often referred to as

dashedness (see, for example, [32]). In this work we also refer to it as the switching state of

a edge or set of edges, motivated by a forthcoming analogy to switching and two-graphs.

Relative to the edge colour ordering, the switching state of an edge e = (u, v) of colour i

will be denoted alternately by πi(u), πi(v) or π(e) (referring to the ith edge of the vertex

u or v, or simply the edge e).

Note that the graph-theoretic definition above corresponds exactly to that provided in

the introductory chapter. The edge-N -partition corresponds to the N supercharges of the

related N -extended superalgebra. The height assignment reflects the relative engineering

148


dimension of the vertices, and the edge parity fixes the sign of the right hand side of Eq.

(7.18), the transformation rule encoded by the edge. Finally, the 4-length cycle property

reflects the condition of Eq. (7.6) that QI , QJ = 0 when I 6= J .

The above conditions imply several additional properties. The edge-N -partite restriction

requires equal numbers of bosons and fermions, hence the bipartition must consist of

two sets of equal size. Together with the 4-length cycle property, the edge-N -partite

requirement also implies that the number of vertices, |V |, be equal to 2n for some n ∈ Z+,

n ≤ N . If n = N , then ignoring edge and vertex colouring, the resulting graph is simply

the N -dimensional hypercube, or N -cube. Hence we term the corresponding Adinkra the∗

N-cube Adinkra. Where n 6= N , we denote the corresponding Adinkra to be an (N, k)

Adinkra, where k = N − n.

When drawing these graphs, we represent the bipartition by black and white vertices.

As in previous work by Doran et al. [27], the height assignments will be represented by

arranging the vertices in rows, incrementally, according to height.

Example 7.2.2. The Adinkras of Fig. 7.1 below satisfy all requirements listed above.

Note that nodes in the same bipartition (bosons or fermions) must be at the same height

modulo 2, and that every 4-length cycle containing only two edge colours has an odd

number of dashed/solid edges. In case (i), this is simply every 4-length cycle, however

case (ii) also contains 4-length cycles with four edge colours.

(i) (ii)

Figure 7.1: (i): A 3-cube Adinkra, and (ii): A (4,1) Adinkra.

It is also worth mentioning at this point that the questions of whether dashed edges corre-

spond to even or odd parity, and which of the black or white node sets correspond to bosons

∗in Section 8.1 we show that this is unique to N

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7.3. ALTERNATIVE REPRESENTATIONS AND MODELS OF ADINKRAS

or fermions, have been left ambiguous, as they do not impact on the following analysis,

and often represent a symmetry in the system. Also note that the absolute height value of

height assignments are also currently ambiguous, as only relative values of height will be

relevant to the following work. Finally, we note that since this work is concerned only with

questions of equivalence and isomorphism of Adinkras, the assumption of connectedness

will be made throughout to simplify the analysis, without loss of generality.

7.3 Alternative Representations and Models of Adinkras

There are other particularly useful ways of defining and modelling Adinkras. Before de-

scribing these, it will be appropriate to introduce some more terminology, specifically

relating to linear codes and Clifford algebras. We must also consider how to define notions

of equivalence and isomorphism between Adinkras.

7.3.1 Equivalence and Isomorphism Definitions

Definition 7.3.1. The topology of an Adinkra is defined as the underlying vertex and

edge sets with all the colourings (including edge parity and height assignments) removed.

When only the colourings associated with edge parity and height assignments are removed,

the resulting graph is termed the chromotopology of the original Adinkra.

Two operations relative to a definition of equivalence have been defined on Adinkras [25].

The vertex lowering / raising operation consists of changing the height of a given set of

vertices while preserving the requirement that adjacent vertices are at adjacent heights.

The operation of switching a vertex consists of reversing the parity of all edges incident

with it. We note briefly that this preserves the property that 4-length cycles with only

two edge colours have an odd number of dashed edges.

This switching operation is analogous to Seidel switching of graphs (see [148,149]) in which

adjacency and non-adjacency is swapped. In this case, adjacency has been replaced by

parity / dashedness for the purposes of switching. Continuing this analogy, we define the

switching class of an Adinkra G to be the set of all Adinkras that can be obtained from

G by switching some subset of its vertices.

Any Adinkras in the same switching class will be considered isomorphic. Adinkras related

via vertex raising / lowering operations will be considered equivalent, but not necessarily

150


isomorphic.

Hence the definitions of equivalence and isomorphism considered here differ slightly. Per-

muting the vertex labels, switching and raising / lowering operations all preserve equiv-

alence, whereas the only operations preserving isomorphism are those of switching and

permutations of vertex labels.

Example 7.3.2. All three Adinkras drawn in Fig. 7.2 are in the same equivalence class,

however only the first two are isomorphic.

Figure 7.2: Three Adinkras belonging to the same equivalence class. The first two areisomorphic, whereas the third belongs to a separate isomorphism class.

An automorphism of an (N, k) Adinkra G(V,E) is a permutation of the vertex set, p :

V → V , p ∈ Sym(2N−k) that is also an isomorphism.

Note that as the ultimate goal of introducing Adinkras is a greater understanding of rep-

resentations of off-shell supermultiplets, useful definitions of equivalence and isomorphism

regarding Adinkras should be related to the underlying supermultiplets that they encode.

However notions of isomorphism applied to supermultiplets are inherently context-specific.

A pertinent discussion relating to automorphisms of supermultiplets can be found in Ap-

pendix B of [132]. Here we cover a few of the points relating directly to Adinkras. Similar

to the notation used in Section 7.1.1, in general supermultiplets can be regarded as a

triple (Q,Rφ,Rψ), where Rφ and Rψ are vector spaces corresponding to the bosonic and

fermionic component fields φ and ψ, and Q = Q1, . . . , QN denotes the set of N super-

charges, or supersymmetry transformation operators mapping between Rφ and Rψ and

their derivatives.

Transformations acting on a supermultiplet are grouped into two classes, inner and outer

transformations. This distinction reflects the property that general supersymmetric mod-

els will consist of several supermultiplets, each consisting of distinct (Rφ,Rψ) pairs whilst

sharing the same set of supercharges Q. An inner transformation is one affecting a single

151


supermultiplet while leaving the others unchanged. For instance permuting the labels of

the component fields of some (Rφ,Rψ) pair affects only that supermultiplet. However a

transformation of the set Q1, . . . , QN affects all supermultiplets of a given supersym-

metric model, and is hence termed an outer transformation.

A transformation induces an automorphism when the result is indistinguishable from the

original; when its image and pre-image are the same. However ‘indistinguishability’ is

inherently contextual, and so there are several notions of equivalence between supermul-

tiplets. Whether an outer transformation induces an automorphism can depend on the

particular model being studied, hence permutations of the Q-labels may or may not pre-

serve equivalence.

Consequently, the definitions of equivalence and isomorphism provided above for Adinkras

are not the only possible definitions, and as such a general classification of Adinkras must

address a broader class of possible equivalences. One such alternative which is consid-

ered in Section 8.3.1 involves the inclusion of edge-colour permutations as an operation

preserving isomorphism. As the edge colours (alternatively known as edge dimensions)

correspond to the supercharges of the underlying supermultiplet, this situation simply

involves allowing permutations of the set Q1, . . . , QN to preserve isomorphism.

To introduce the tools necessary to classify the automorphism group properties of

Adinkras, it will first be necessary to give a brief description of linear codes and Clif-

ford algebras.

7.3.2 Linear Codes

A binary linear (N, k) code C consists of a set of 2k codewords (bit-strings) of length N

forming a group under their bitwise sum modulo 2 (exclusive or), represented by . Hence

for all u, v ∈ C, u v ∈ C. When dealing with a bit-string or codeword u of length N we

will use the notation u = (u1, u2, . . . , uN ), where ui is the ith bit of u. We also consider all

subsequent codes to be both binary and linear; these terms are henceforth dropped from

their description.

A generating set of C is a set of k codewords D = (c1, . . . , ck) ⊂ C such that all codewords

in C can be formed by a linear combination of elements of D. The value k is called the

dimension of the code, and is invariant with respect to the particular choice of generating

set.

152


The weight of a codeword u, denoted by wt(u), or simply |u|, is the number of 1’s in u. If

every codeword in a code has a weight of 0 (mod 4) the code is called doubly even. The

inner product of two codewords u and v of length N is defined as

〈u, v〉 ≡N∑

i=1

ui vi (mod 2).

Then a doubly even code has the property that any two of its codewords have an inner

product of 0.

Definition 7.3.3. A standard form generating set D of an (N, k) code C is defined to

be of the form D = (I | A), where I is the k × k identity matrix and A is a k × (N − k)

matrix being the remainder of each codeword in D.

Example 7.3.4. By applying a permutation of the bit-string order and considering an

alternative generating set, the code d8, generated by

(1 1 1 1 0 0 0 00 0 1 1 1 1 0 00 0 0 0 1 1 1 1

)can be written in

terms of a standard form generating set

(1 0 0 1 1 1 0 00 1 0 1 1 0 1 00 0 1 1 1 0 0 1

). Note that generating sets will be

assumed to be minimal, in that the codewords comprising the set are linearly independent.

Thus, a generating set with n codewords corresponds to a code with 2n distinct codewords.

Observation 7.3.5. For any doubly even (N, k) code, there exists a generating set D and

permutation of the bit-string order p ∈ Sym(N) such that D is a standard form generating

set.

The notion of bit-strings can be effectively used to furnish a further organisation of the

edge and vertex sets of an Adinkra. If we order the edge colours of an N -cube Adinkra

from 1 to N , then assign each of these edge colours to the corresponding position in an

N -length bit-string, each of the 2N vertices in the Adinkra can be represented by such a

bit-string. These bit-strings are allocated in such a way that two vertices connected by

the ith edge differ precisely in the ith element of their respective bit-strings.

Example 7.3.6. Applying this bit-string representation to the vertices of a 3-cube

Adinkra yields the graph of Fig. 7.3 below. Note that for an N -cube Adinkra, two vertices

are connected by an edge if and only if their corresponding bit-string representations have

a Hamming distance of 1 (differing in only one position).

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Figure 7.3: A 3-cube Adinkra shown together with a bit-string representation for eachvertex.

An important theorem of [26] links general (N, k) Adinkras to double even codes. We

paraphrase it, combined with other results from the same work, below.

Theorem 7.3.7 ( [26]). Every (N, k) Adinkra (up to equivalence) class can be formed

from the N -cube Adinkra by the process of quotienting via some doubly even (N, k) code.

This quotienting process works as follows. We start with a doubly even (N, k) code C and

an N -cube Adinkra with a corresponding edge colour ordering and bit-string representa-

tion. We then identify vertices (and their corresponding edges) related via any codeword

in C, ensuring first that identified vertices have identical edge parities relative to the or-

dering of edge colours. In this process an (N, k) code identifies groups of 2k vertices, and

reduces the order of the vertex set from 2N to 2N−k, producing an (N, k) Adinkra.

7.3.3 Clifford Algebras

Definition 7.3.8. We will consider the Clifford algebra Cl(n) to be the algebra over Z2

with n multiplicative generators γ1, γ2, . . . , γn, with the property

γi, γj = 2δij1, (7.20)

where δij is the Kronecker delta. In other words, each of the generators is a root of 1, and

any two generators anticommute.

It will be convenient to view the 4-length cycle condition of Adinkras in terms of the

154


anticommutativity property of Clifford generators. If we consider each position in the bit-

string representation of a vertex of an N -cube Adinkra to correspond to a given generator

of Cl(N), with the vertex itself being related to the product of these Clifford generators,

then the ith edge dimension can be naturally viewed as the transformations corresponding

to left Clifford multiplication by the ith Clifford generator.

For example, a vertex with bit-string representation (01100) corresponds to the product

of Clifford generators γ2γ3 ∈ Cl(5), and the edge connecting the vertex (01100) to (01101)

is related to the mapping γ2γ3 7→ (γ5)(γ2γ3) = γ2γ3γ5. Then the condition that every

4-length cycle containing exactly two edge colours has an odd number of dashed edges

corresponds to the anticommutativity property of Clifford generators, γiγj = −γjγi for

i 6= j.

In connection with this Clifford generator notation, we define a standard form for the

switching state of an N -cube Adinkra.

Definition 7.3.9. A standard form N -cube Adinkra has edge parity as follows. The ith

edge of vertex v = (v1, v2, . . . , vN ) has edge parity given by

πi(v) ≡ (v1 + v2 + . . .+ vi−1) (mod 2). (7.21)

This corresponds to left Clifford multiplication of v by γi. Note that this standard form

is defined relative to some given bit-string labelling of the vertex set. Since fixing the

labelling of a single vertex fixes that of all vertices, we will sometimes refer to a standard

form as being relative to a source node, being the vertex with label (00 . . . 0), or simply

standard form relative to 0.

Example 7.3.10. The 3-cube Adinkra of Example 7.3.6 is in standard form. Vertices are

ordered into heights relative to their bit-strings, such that heights range from 0 (at the

bottom) to 3, and vertices at height i have weight i. The edges are ordered from left to

right, such that green corresponds to γ1 and red corresponds to γ3.

As we have seen, many† Adinkras can be formed by quotienting the N -cube Adinkra

with respect to some doubly even code. However in practice this method can be quite

inefficient. The following alternative method for constructing an (N, k) Adinkra, with

associated code C is due to G. Landweber (personal communication), and is used by the

Adinkramat software package:

†As we have not yet investigated the case of Gnomon Adinkras defined by the ‘zippering’ processpresented in [150], the extension of our results to these cases requires further study.

155


1. Start with the standard form (N − k)-cube Adinkra induced on the first N − k edge

dimensions.

2. Find a standard form generating set for C, of the form D = (I | A).

3. Associate the (N − k + i)th edge dimension with the product of Clifford generators

given by Ai, the ith row of A.

4. The ith edge connects a vertex v to the vertex v.Ai, with switching state correspond-

ing to right Clifford multiplication of v by Ai, up to a factor of (−1)F .

The factor of (−1)F , termed the fermion number operator, simply applies a (−1) factor to

fermions, and leaves bosons unchanged. This factor is required to ensure the parity of an

edge remains the same in either direction. Again, note that the choice of whether even-

or odd-weight vertices are bosons remains ambiguous. For simplicity, we consider odd-

weight vertices to be fermions for the purposes of the (−1)F factor, with the symmetry

encoded by a graph-wide factor of ±1 in the switching state of the extra k edge dimensions,

corresponding to two potentially inequivalent (N, k) Adinkras.

It remains to be seen whether this construction method yields all (N, k) Adinkras. To our

knowledge this has not been explicitly proven in previous work, hence we present a proof

of this as follows.

Theorem 7.3.11. All (N, k) Adinkras of a given equivalence class can be formed from

the (N − k)-cube Adinkra by the construction method detailed above.

Proof. As we are only considering equivalence classes, the height assignments will be

ignored.

Consider an N -cube Adinkra quotiented with respect to an (N, k) code with standard form

generating set D = (I | A), producing an (N, k) Adinkra G(V,E). Assume without loss of

generality that the vertices remaining after quotienting all have fixed (N −k+1)th → N th

bit-string characters (for instance all fixed as 0). Then the rows of A represent the vertices

identified by the quotienting process. We want to show that, up to isomorphism, there

are at most two different ways this quotienting operation can be performed.

Isolating any set of N − k edge dimensions of G yields an induced (N − k)-cube Adinkra,

H(V,E′), with E′ ⊂ E. We will see in Section 8.1 that for fixed m, all m-cube Adinkras

belong to a single equivalence class. Hence without loss of generality H can be considered

156


to be in standard form. This in turn fixes the switching state of all edges belonging to

the chosen N − k edge dimensions. Now only a single degree of freedom remains in the

switching states of the remaining k edge dimensions, due to the 4-length cycle condition.

In other words, fixing the switching state of any one of the remaining edges fixes all

remaining edges. This choice corresponds to a graph-wide factor of ±1 in the switching

state of the additional k edge dimensions.

Since this graph-wide factor of ±1 matches the difference between the two possibly in-

equivalent Adinkras produced by the above construction method, it remains to show that

the construction method does indeed produce valid (N, k) Adinkras. Hence me must verify

that the 4-length cycle condition holds.

As odd-weight vertices are considered to be fermions, the (−1)F factor can be replaced

by an additive factor of∑N−k

j=1 xj , for a given vertex x ∈ V . Then for i > N − k, x ∈ V ,

Ai = (a1, a2, . . . , aN−k),

πi(x) ≡N−k∑

j=1

xj + x . Ai (7.22)

≡N−k∑

j=1

xj + x2(a1) + x3(a1 + a2) + . . .+ xN−k(a1 + a2 + . . .+ a(N−k−1)) (mod 2).

The 4-length cycles in G with two edge colours i and j will be split into three cases:

(i) i, j ≤ N − k, i < j

(ii) i ≤ N − k, j > N − k and

(iii) i, j > N − k.

Note that a given vertex x ∈ V defines a unique such 4-length cycle. In case (i), consider

two antipodal points of such a 4-length cycle, x and x+ i+ j. Then the sum of the edge

parities of the 4-length cycle equals:

(x1 + . . .+ xi−1) + (x1 + . . .+ xj−1) +

(x1 + . . .+ xi−1) + (x1 + . . .+ (xj + 1) + . . .+ xj−1) ≡ 1 (mod 2). (7.23)

Hence the 4-length cycle condition holds (note that this is the only case present in the

construction of the standard form N -cube Adinkra).

157


For case (ii), the antipodal points are x and z, where

z ≡ (x1 + a1, . . . , xi + ai + 1, . . . , xN−k + aN−k)

≡ x+Aj + i (mod 2), (7.24)

and by substituting Eq. 7.22, the sum becomes:

(x1 + . . .+ xi−1) + (N−k∑

l=1

xl + x.Aj) + (x1 + . . .+ xi−1) +

(a1 + . . .+ ai−1) + (N−k∑

l=1

zl + z.Aj) (mod 2). (7.25)

However |Aj | ≡ 3 (mod 4), so |x| ≡ |z| (mod 2). Then the first and third terms cancel,

as do the second and fifth, leaving:

x.A + z.A + (a1 + . . .+ ai−1)

≡ 2(x.A) + 2(a1 + . . .+ ai−1) + a2(a1) + a3(a1 + a2)

+ . . .+ aN−k(a1 + . . . a(N−k−1))

≡ |Aj − 1|+ |Aj − 2|+ . . .+ 1

≡ 1

2|A− 1||A| (mod 2). (7.26)

Now |A| ≡ 3 (mod 4), hence

1

2|A− 1||A| ≡ 3 (mod 4)

≡ 1 (mod 2), (7.27)

and case (ii) is verified.

Case (iii) proceeds similarly to case (ii).

Hence the graphs produced via this construction method are indeed (N, k) Adinkras.

Since [26] showed that all (N, k) Adinkras can be produced via the quotienting method,

there can be at most two equivalence classes of Adinkras with the same associated code,

and hence the Adinkras produced from the two methods must coincide.

Definition 7.3.12. A standard form (N, k) Adinkra has switching state given by the

above construction, relative to an associated doubly even code.

158


Example 7.3.13. The smallest non-trivial (N, k) Adinkra has parameters N = 4, k = 1,

with associated doubly even code (1111). The standard form of Definition 7.3.12 is shown

in Fig. 7.4 below, for each choice of the graph-wide factor of ±1.

(000) (001)

(000) (001) (001) (010)

(010) (011)

(011) (100) (101)

(100) (101) (101) (110)

(110) (111)

(000) (001)

(000) (001) (001) (010)

(010) (011)

(011) (100) (101)

(100) (101) (101) (110)

(110) (111)

Figure 7.4: Two (4,1) Adinkras with switching state in standard form.

In this case the two Adinkras formed belong to different equivalence classes, as we demon-

strate in the following section.

This construction provides a much simpler practical method of constructing Adinkras,

and the standard form will be convenient in the proof of later results. Note that all (N, k)

Adinkras have an associated doubly even code, with matching parameters of length and

dimension. For these purposes the N -cube Adinkra is considered an (N, 0) Adinkra with

a trivial associated code of dimension 0.

159

Chapter 8

Classification of Adinkras

8.1 Automorphism Group Properties of Adinkras

We now have sufficient tools to derive the main result of this Part: classifying the auto-

morphism group of an Adinkra with respect to the local parameters of the graph, namely

the associated doubly even code. We start with several observations regarding the auto-

morphism properties of Adinkras.

Lemma 8.1.1. Ignoring height assignments†, the N -cube Adinkra is unique to N , up to

isomorphism.

Proof. It suffices to show that any N -cube Adinkra can be mapped to standard form via

some set of vertex switching operations. Such a mapping can be trivially constructed.

For example, start with a single vertex. The parity of its edge set can be mapped to any

desired form by a set of vertex switchings applied only to its neighbours. In particular

the edge parities can be mapped to those of the corresponding standard form Adinkra.

Since the 4-length cycle restriction of Section 7.2.2 holds, this process can be continued

consistently for the set of vertices at each subsequent distance from this original vertex.

The Adinkra resulting from this process will be in standard form relative to this original

vertex, regardless of the initial switching state of the Adinkra.

†Equivalently, there are only two heights, corresponding to bosons and fermions. Adinkras withthis property are called valise Adinkras. Adinkras with more than two heights are termednon-valise.

160

CHAPTER 8. CLASSIFICATION OF ADINKRAS

Lemma 8.1.2. N -cube valise Adinkras are vertex-transitive up to the boson/fermion bi-

partition.

Proof. To show this, we begin with a standard form N -cube Adinkra G(V,E), with switch-

ing state defined by left multiplication by the corresponding Clifford generators, as in

Definition 7.3.9. Since N -cube Adinkras are unique up to isomorphism, in the sense of

Lemma 8.1.1, this can be done without loss of generality.

Then the switching state of edge (x, y), where x = i.y, is of the form x1 + x2 + . . .+ xi−1

(mod 2). Consider a non-trivial permutation ρ : V 7→ V . Since ρ is non-trivial, there are

two vertices u, v in V, u 6= v, such that ρ : u 7→ v. If ρ is to preserve the topology of G,

we must have ρ : x 7→ (u.v)x, ∀x ∈ V . Note that ρ has no fixed points, and is completely

defined in terms of (u.v). Any automorphism γ of G which permutes V according to ρ

must also switch some set of vertices in such a way that edge parity is preserved. However

ρ maps the edge (x, y), where x = i.y, with edge parity x1 + x2 + . . . + xi−1 (mod 2), to

the edge (ρ(x), ρ(y)), with edge parity of

(x1 + x2 + . . .+ xi−1) + ((u1 + v1) + (u2 + v + 2) + . . .+ (ui−1 + vi−1)) (mod 2). (8.1)

Hence the change in edge parity of (x, y) does not depend on either endpoint explicitly,

only on the mapping ρ and the edge dimension i. In other words, ρ either preserves the

switching state of all edges of a given edge dimension, or reverses the parity of all such

edges. Note that any single edge dimension can be switched (while leaving all other edges

unchanged) via a set of vertex switching operations, hence an automorphism γ exists for

all such ρ. Furthermore, γ is unique to ρ, which is in turn unique to the choice of (u.v).

Note that if G is in standard form, we term the automorphism mapping vertex x to vertex

0 = (00 . . . 0) as putting G in standard form relative to x. By this terminology, G was

initially in standard form relative to 0.

Corollary 8.1.3. The N -cube Adinkra is minimally vertex-transitive (up to the biparti-

tion), in the sense that the pointwise stabiliser of the automorphism group is the identity,

and |Aut(G)| = 12 |V | = 2N−1. In other words, no automorphisms exist that fix any points

of the N -cube Adinkra.

Consider any (N, k) Adinkra G(V,E). The sub-Adinrka H(V,E′) induced on any set S

of (N − k) edge dimensions of G will be an (N − k)-cube Adinkra. Also, for all such S

161

8.1. AUTOMORPHISM GROUP PROPERTIES OF ADINKRAS

there exists some set of vertex switching operations such that the induced Adinkra H is

in standard form. In the case where S = (1, 2, . . . , N − k), the induced H is in standard

form if and only if G is in standard form.

Observation 8.1.4. Given some (N, k) Adinkra, containing a standard form (N − k)-

cube Adinkra induced on the first (N − k) edge dimensions (or equivalently an induced

(N − k)-cube Adinkra in any given form), the switching state of the ith edge dimension,

i > (N − k), is fixed up to a graph wide factor of −1.

In particular, fixing the parity of the extra edges i, where i > (N − k), of a single ver-

tex fixes the parity of all extra edges in the graph. This is a direct consequence of the

anticommutativity property of 4-length cycles with two edge colours.

This reduces the problem of finding automorphisms of a general (N, k) Adinkra to that of

local mappings between vertices. In particular, for any two nodes x, y ∈ V , there will be

a unique automorphism of the induced (N − k)-cube Adinkra mapping x to y. This will

extend to a full automorphism of G if and only if it preserves the switching state of the

additional k edges of x. Hence we arrive at the following result.

Theorem 8.1.5. Consider an (N, k) Adinkra G(V,E) having an associated code C with

standard form generating set D = (I | A). Two vertices x, y in V are equivalent (disre-

garding vertex colouring, there is an automorphism mapping between them) if and only if

their relative inner products with respect to each codeword in D are equal. In other words,

∃ γ ∈ Aut(G) such that γ : x 7→ y iff ∀ c ∈ D, 〈x, c〉 ≡ 〈y, c〉 (mod 2).

Proof. Consider without loss of generality the case where G is in standard form, and take

H(V,E′) to be the sub-Adinkra induced on the first N − k edge dimensions. By Lemma

8.1.2, H is vertex transitive if we disregard vertex colouring. So consider the automorphism

ρ of H, ρ : x 7→ y, where x, y ∈ V . Now ρ consists of two parts: A permutation of the

vertex set, ρ : v 7→ x.y.v,∀ v ∈ V , and a set of vertex switching operations, such that the

switching state of H is preserved.

We wish to know when ρ switches x relative to y (i.e. when x is switched but y is not, or vice

versa, and when either both or neither are switched). Where πi(x) denotes the switching

state of the ith edge of x (in G or H), we use π′i(x) to denote the equivalent switching

state of x in ρ(G) or ρ(H) (we also denote ρ(G) and ρ(H) by G′ and H ′ respectively. Note

that x in G′ is the image of y in G under ρ.

162


For edge dimension i ≤ N−k, we see that since ρ is an automorphism of H, πi(x) = π′i(y),

and similarly πi(v.x) = π′i(v.y) ∀ v ∈ V . For the case i > N − k, πi(v) is given by Eq.

7.22 in the proof of Theorem 7.3.11, but what about π′i(v)?

The value of π′i(v) can be found by considering the path Pi(x) in H joining x to x.Ai, where

Ai = (a1, a2, . . . , aN−k). If we denote by Ai the associated product of Clifford generators

Ai =n∏

j=1

γsj ,

where the sj denote the 1’s of Ai, we have:

Pi(x) = (x, x.as1 , (x.as1).as2 , . . . , x.Ai).

Then ρ will preserve the parity of edge (x, x.Ai) if and only if the number of dashed edges

in paths Pi(x) and Pi(y) are equal (mod 2). In other words,

πi(x) = π′i(x) iff∑

e∈Pi(x)

π(e) ≡∑

e∈Pi(y)

π(e) (mod 2). (8.2)

Now πs1(x) ≡ x1 + . . .+ xs1−1 (mod 2), . . . , πsn(x.Ai) ≡ (x1a1) + . . .+ (xsnasn) (mod 2),

where ai = 1 for i ∈ sj : 1 ≤ j ≤ n, and ai = 0 elsewhere. Hence this can be rearranged

as

∑

e∈Pi(x)

π(e) ≡ a2(x1) + a3(x1 + x2) + . . .+ aN−k(x1 + . . .+ xN−k−1)

≡ x1(a2 + . . .+ aN−k) + . . .+ xN−k−1(aN−k) (mod 2). (8.3)

Hence, πi(x) ≡ |x| + x.Ai (mod 2), and π′i(y) ≡ πi(x) + (∑

e∈Pi(x) π(e) −∑e∈Pi(y) π(e)),

and note that∑

e∈Pi(x) π(e) + x.Ai simplifies to

|Ai||x| −N−k∑

j=1

(xjaj), (8.4)

so we have, for i > N − k,

πi(x)− π′i(y) ≡ |x|+ |Ai||x|+ |y|+ |Ai||y|+N−k∑

j=1

(xjaj) +

N−k∑

j=1

(yjaj) (mod 2). (8.5)

Since |Ai| ≡ 1 (mod 2), we have |x|(|Ai| + 1) ≡ 0 (mod 2), and the first 4 terms cancel,

163

8.1. AUTOMORPHISM GROUP PROPERTIES OF ADINKRAS

leaving

πi(x)− π′i(y) ≡N−k∑

j=1

(xjaj) +N−k∑

j=1

(yjaj)

≡ 〈x,Ai〉+ 〈y,Ai〉 (mod 2), (8.6)

which completes the proof.

Observation 8.1.6. Two vertices having the same relative inner products with respect

to each element of a set S of codewords also have the same relative inner product with

respect to the group generated by S, under bitwise addition modulo 2.

Corollary 8.1.7. Conversely to Theorem 8.1.5, consider a single orbit φ of the (N, k)

Adinkra G with associated doubly even code C. If all elements of φ have fixed inner product

relative to an N -length codeword c, then c ∈ C, the code by which G has been quotiented.

Theorem 8.1.5 leads to several important corollaries regarding the automorphism group

properties of Adinkras. Recall that there are at most two equivalence classes of Adinkras

with the same associated doubly even code. Theorem 8.1.5 implies that, disregarding the

vertex colourings, there is in fact only one such equivalence class. Hence including the

vertex bipartition, we see that two equivalence classes exist if and only if the respective

vertex sets of bosons and fermions are setwise non-isomorphic. This in turn occurs if and

only if at least one orbit (and hence all orbits) of G is of fixed weight modulo 2.

In [26] and [32], one such case was investigated, for Adinkras with parametersN = 4, k = 1.

In this case, the Klein flip operation, which exchanges bosons for fermions, was found in [26]

and [32] to change the equivalence class of the resulting Adinkra. We term this property

Klein flip degeneracy, and note the following corollary of Theorem 8.1.5.

Corollary 8.1.8. An (N, k) Adinkra with associated code C has Klein flip degeneracy if

and only if the all-1 codeword (11 . . . 1) ∈ C. This in turn occurs only if N ≡ 0 (mod 4).

Corollary 8.1.9. The automorphism group of a valise (N, k) Adinkra G(V,E) has size

2N−2k−a, with 2k+a orbits of equal size, where a = 0 if C contains the all-1 codeword, and

a = 1 otherwise.

Proof. G has 2N−k nodes. By Theorem 8.1.5, disregarding the vertex bipartition there are

2k orbits of equal size, partitioning the vertex set. Including the bipartition, these orbits

164


are split in 2 once more whenever they contain nodes of variable weight modulo 2 (i.e.

whenever C does not contain the all-1 codeword).

8.2 Characterising Adinkra Degeneracy

In the work of Gates et al. [32] the Klein flip degeneracy in the N = 4 case of Example

7.3.13 was investigated. It was shown that Adinkras belonging to these two equivalence

classes can be distinguished via the trace of a particular matrix derived from each Adinkra.

We will briefly introduce these results, and generalise them to general (N, k) Adinkras.

Throughout the following section we will consider an (N, k) Adinkra G(V,E) with corre-

sponding code C. Note that G has 2N−k vertices, each of degree N .

8.2.1 Notation

Definition 8.2.1. The adjacency matrix A(G) (or simply A where the relevant Adinkra

is clear from the context) is a 2N−k × 2N−k symmetric matrix containing all the infor-

mation of the graphical representation, except the vertex colouring associated with height

assignments. Each element of the main diagonal represents either a boson or a fermion,

denoted by +i for bosons and −i for fermions. The off-diagonal elements represent edges

of G, numbered according to edge dimension, with sign denoting dashedness. For ex-

ample, if position Ax,y = −4, there is a dashed edge of the fourth edge colour between

boson/fermion x and fermion/boson y. Hence the sign of the main diagonal elements rep-

resents the vertex bipartition, the sign of off-diagonal elements represents the edge parity,

the absolute value of off-diagonal elements represents edge colour, and the position of the

elements encodes the topology of the Adinkra.

Example 8.2.2. Consider the 3-cube Adinkra of Example 7.3.6.

If we order the vertex set into bosons and fermions, this Adinkra can be represented by

165

8.2. CHARACTERISING ADINKRA DEGENERACY

the matrix:

A =

i 0 0 0 1 2 3 0

0 i 0 0 −2 1 0 3

0 0 i 0 −3 0 1 −2

0 0 0 i 0 −3 2 1

1 −2 −3 0 −i 0 0 0

2 1 0 −3 0 −i 0 0

3 0 1 2 0 0 −i 0

0 3 −2 1 0 0 0 −i

. (8.7)

Note that the symmetric nature of the adjacency matrix arises naturally out of the defi-

nition, and requiring that this property be upheld imposes no further restrictions in and

of itself.

We define L and R matrices similarly, as in [32], to represent a single edge dimension of

the Adinkra. L and R matrices encode this edge dimension, with rows corresponding to

bosons and fermions respectively. As each edge dimension is assigned a separate matrix,

the elements corresponding to edges are all set to ±1. Then the Adinkra of Example 8.2.2

has L matrices:

L1 =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

, L2 =

0 1 0 0

−1 0 0 0

0 0 0 −1

0 0 1 0

, L3 =

0 0 1 0

0 0 0 1

−1 0 0 0

0 −1 0 0

. (8.8)

These matrices can be read directly off the top right quadrant of the adjacency matrix.

Similarly, the R matrices can be read off the lower left quadrant. In this case, we have

Ri = LTi , where T denotes the matrix transpose. Note that L and R matrices taken directly

from the adjacency matrix will always be related via matrix transpose. Furthermore, since

we are considering only off-shell supermultiplets, these matrices must also be square. We

will assume that all subsequent L and R matrices are related in this way. We define a

further object, γi, to be a composition of L and R matrices of the form

γi =

0 Li

Ri 0

. (8.9)

166


Further details regarding these objects can be found in [32]. In particular, it was shown

in [32] that the Klein flip degeneracy of several (4, 1) Adinkras can be characterised by

the trace of quartic products of these matrices, according to the formula

Tr(Li (Lj)TLk (Ll)

T) = 4 (δijδkl − δikδjl + δilδjk + χ0εijkl) , (8.10)

where δ represents the Kronecker delta and ε the Levi-Civita symbol, and where χ0 =

±1 distinguishes between the (4, 1) Adinkras belonging to the two equivalence classes of

Example 7.3.13. It is also conjectured in [32] that this same method for distinguishing

equivalence classes can be generalised to all values of the parameters N and k. We prove

this conjecture in the following section.

8.2.2 Degeneracy for General N

In order to generalise Eq. 8.10 to higher N , we wish to determine the form taken by the

trace of products of these γ matrices, and establish exactly when this can be used to

partition Adinkras into their equivalence classes. Firstly we note the following properties

of Adinkras:

Lemma 8.2.3. The cycles of an N -cube Adinkra consist entirely of paths containing each

edge dimension 0 times (modulo 2).

Lemma 8.2.4. The cycles of an (N, k) Adinkra with associated code C consist of paths

in which:

(i) Each edge dimension is traversed 0 (mod 2) times.

(ii) At least one edge dimension is traversed 1 (mod 2) times.

In case (ii), the set of such edge dimensions traversed 1 (mod 2) times corresponds to a

codeword in C.

In fact, the codeword (00 . . . 0) is in any such code C, so all cycles have this property,

however case (i) is considered a trivial instance.

Consider the product of t γ matrices of G, denoted by M = γi1γi2 ...γit . As we are

considering only L and R matrices such that Li = RTi , all γ matrices defined as in Eq. 8.9

will be real and symmetric. This leads to the following property for M .

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8.2. CHARACTERISING ADINKRA DEGENERACY

Lemma 8.2.5. The elements of the main diagonal of M are non-zero if and only if the

path PM = (i1, i2, . . . , it) represents a closed loop within the Adinkra. This can occur one

of two ways:

(i) Trivially, if each edge dimension contained in p is present 0 (mod 2) times,

(ii) If p corresponds to a codeword (or set of codewords) in C.

Proof. Consider the case PM = (i, j). If i = j, M is trivially the identity, hence (i) holds.

If i 6= j, then Mx,y 6= 0 if and only if the vertices x and y are connected via a path (i, j)

(i.e. if x = i.j.y). Extending to general M , if Mx,x 6= 0 for some x ∈ V , this implies that

x is connected to itself via the path PM , a closed loop / cycle in G. Hence Lemma 8.2.4

completes the proof.

Corollary 8.2.6. Mx,x 6= 0 for some x ∈ V if and only if this is true for all x ∈ V .

Lemma 8.2.7. In case (ii) of Lemma 8.2.5, with Mx,x = ±1 for all x ∈ V , Mx,x = My,y if

and only if x and y have the same inner product with respect to the codeword corresponding

to PM .

Proof. Consider the product of Clifford generators corresponding to the path PM . Since

(ii) holds, after cancelling repeated elements we are left with some codeword p ∈ C, such

that p = aPM , where a = ±1, depending on whether PM corresponds to an even or odd

permutation of p, relative to shifting and cancelling of Clifford generators. Since C is a

group, either

(i) p ∈ D, a standard form generating set of C, or

(ii) p = (g1 g2 . . . gr), where gi ∈ D, for any such D.

Assume (i) holds. Then p = (p1, p2, . . . , pt) such that pi ≤ (N − k) for i 6= t, and

pt > (N − k), with the first (N − k) edge dimensions defined relative to the particular

standard form generating set D being considered. In other words, pt = p1 · p2 · . . . · pt−1.

Then ∀x ∈ G, x = (x1x2 . . . xN−k), and the sign of Mx,x, omitting a factor of a−12 , is given

168


by (substituting the formulas for πi(x) of Section 7.3)

(p1 · x+ p2 · x+ . . .+ pt−1 · x) + x · pt≡ p2(x1) + p3(x1 + x2) + . . .+ pt−1(x1 + x2 + . . .+ xt−2)

+ p1(x2 + . . .+ xt−1) + . . .+ pt−2(xt−1) + |x|

≡ (|p| − 1)|x| −t−1∑

i=1

(pixi) + |x|

≡ |x||p|+ 〈x, p〉 (mod 2). (8.11)

Note that |p| ≡ 0 (mod 4), so the first term disappears. The same arguments can be

followed to show that this holds for case (ii) also. Then since the factor of a−12 is constant

for all x ∈ G, it follows that for all x, y ∈ G, Mx,x = My,y if and only if 〈x, p〉 = 〈y, p〉.

One immediate corollary of the preceding lemma is that the trace of M will vanish when-

ever PM corresponds to a codeword in C. In fact, Tr(M) will only be non-zero in the

trivial case where the path PM consists of a set of pairs of edges of the same colour. These

are the paths corresponding to the Kronecker delta terms of Eq. 8.10. In particular, this

implies that Tr(M) cannot distinguish between equivalence classes of Adinkras directly.

However if we instead consider powers of L and R matrices, the preceding lemma suggests

a direct generalisation of Eq. 8.10 to all (N, k) Adinkras.

Given an (N, k) Adinkra, consider the product of t L matrices, Li1LTi2 . . .L

Tit . Denoting

the path p = (i1, i2, . . . , it), we define the value σp such that σp = 0 if there exists an

edge dimension in p that is present 1 (mod 2) times, and σp = a otherwise. Here a = ±1,

corresponding to the sign of the related product of Clifford generators (since p consists of

pairs of edges of the same colour, this product of Clifford generators equals ±1). Then we

have the following result.

Lemma 8.2.8. The trace of Li1LTi2 . . .LiN−1LT

iN, where p = (i1, i2, . . . , iN ), equals

2N−k−1 (σp + χ0εp), (8.12)

where χ0 = ±1 depending on the equivalence class of the (N, k) valise Adinkras.

Proof. The σp term follows directly from Lemmas 8.2.5 and 8.2.7. In Section 8.1, we show

that (N, k) Adinkras with the same associated code C are all in a single equivalence class,

169

8.3. IDENTIFYING ISOMORPHISM CLASSES OF ADINKRAS

except where (11 . . . 1) ∈ C. In this case, the two equivalence classes are related via the

Klein flip operation, exchanging bosons and fermions. Moreover, since c = (11 . . . 1) ∈ C,

the Klein flip operation switches the sign of 〈x, c〉, for each x ∈ V . Then by Lemma

8.2.7, the Adinkras from different equivalence classes correspond to χ0 values of opposite

sign.

Note that if N = 4 and p = (i, j, k, l), then σp = δijδkl−δikδjl+δilδjk, and the result of Eq.

8.10 follows. These results can be simplified in some respects if we consider the products

of γ matrices instead. Recall the definition M = γi1γi2 ...γit , where PM = (i1, i2, . . . , it),

and the fermion number operator (−1)F . Instead of taking the trace of M , consider the

trace of M.(−1)F. We have

γi =

0 Li

Ri 0

and (−1)F =

1 0

0 −1

, (8.13)

and we are considering only L = RT, so for even t,

M =

Li1LT

i2 . . .LTit 0

0 LTi1Li2 . . .Lit

. (8.14)

Replacing L matrices by R matrices in Lemma 8.2.8 simply changes the sign of χ0 , so for

an (N, k) Adinkra,

Tr(M · (−1)F) = 2N−k χ0εp. (8.15)

8.3 Identifying Isomorphism Classes of Adinkras

The results up to this point deal with equivalence classes of valise Adinkras, where we

consider only 2-level Adinkras. For the non-valise case, we require a method of parti-

tioning general (N, k) Adinkras into their isomorphism classes. The essential problem in

establishing such a method is in classifying the topology of an Adinkra relative to the au-

tomorphism group of its underlying 1-level Adinkra (in which vertex colouring is ignored).

We might consider simply partitioning the vertex set into the orbits of this underlying

automorphism group, and then classifying each height by the number of vertices of each

orbit that it possesses. However this method will clearly be insufficient, as it ignores the

relative connectivity between vertices at different heights. For example, the two Adinkras

170


of Fig. 8.1 below are in the same equivalence class by use of this set of definitions. They

also have the same number of vertices from each orbit at each height, and yet they are

clearly non-isomorphic; no relabelling of the vertices or vertex switching operations can

map between them. However even this simple method does come close to partitioning

Figure 8.1: Two equivalent, non-isomorphic Adinkras with the same values of µ(h, a), forall possible values of h and a.

non-valise Adinkras into their isomorphism classes. Recall that a pointwise stabiliser of

the automorphism group of any Adinkra is the identity - no non-trivial automorphisms

exist that fix any vertex†. In fact fixing any vertex of an Adinkra yields a natural, canon-

ical ordering of the vertex set, relative to some ordering of the edge dimensions, according

to the following construction.

Construction 8.3.1. Suppose we are given an (N, k) Adinkra G(V,E), together with an

ordering of the edge dimensions of (i1, i2, . . . , iN ). Then fix (choose) any vertex v ∈ V .

We define an ordering λv of the vertex set relative to v, where λv : V 7→[2N−k

], in the

following way.

• λv(v) = 1.

• Order the neighbours of v from 2 to N + 1 according to the ordering of the corre-

sponding edge dimensions (λv(inv) = 1 + n, where 1 ≤ n ≤ N).

• Repeat this for vertices at distance 2, beginning with the neighbours of i1v, and

ending with the neighbours of iNv.

• Repeat this similarly for vertices at each distance, until all vertices have been as-

signed an ordering in[2N−k

].

†Here the term non-trivial refers to the corresponding permutation of the vertex set. An automorphismwhich switches vertices but leaves the ordering of the vertex set unchanged is considered trivial.

171


In other words, λv firstly orders the vertex set according to distance from v, then for each

of these sets, each vertex x is assigned an ordering based on the lexicographically smallest

path from v to x.

To formalise the partitioning of heights discussed above, consider an (N, k) Adinkra

G(V,E) with h different height assignments. Let ΓG be the automorphism group of the

corresponding 1-level Adinkra (ignoring the vertex colouring of G). Relative to a partic-

ular ordering of the edge dimensions, and a particular generating set of the associated

doubly even code, order the 2k orbits of ΓG according to τ : V 7→[2k]. We then define

µG(h, a) to be the number of vertices at height h belonging to the ath orbit of ΓG, such

that

µG(h, a) = |v ∈ V : hgt(v) = h, τ(v) = a|. (8.16)

Consider a pair of Adinkras G(V,E) and H(V,E′) belonging to the same equivalence

class, both having t distinct heights. Then G and H have the same associated code C.

If µG(h, a) = µH(h, a), for all 1 ≤ h ≤ t, relative to a given edge-colour ordering and

generating set of C, then choose any vertex v from each Adinkra belonging to the same

orbit and height. Relative to this vertex v, consider the unordered set

certG(v) = (λv(x), hgt(x), τ(x)) : x ∈ V . (8.17)

Theorem 8.3.2. Consider two Adinkras G and H and a vertex v from each with the prop-

erties described above. Then certG(v) = certH(v) if and only if G and H are isomorphic.

Proof. If G and H are isomorphic then this is trivially true. Conversely, assume

certG(v) = certH(v). Then since G and H belong to the same equivalence class, and

τG(v) = τH(v), there exists an isomorphism γ mapping G to H (ignoring the vertex

colourings). Then γ extends to a full isomorphism (including the vertex colourings) if it

preserves height assignments. This follows directly from certG(v) = certH(v), hence G

and H are isomorphic.

The preceding theorem provides an efficient method of classifying Adinkras according to

their isomorphism class. Note that the classification is relative to a given ordering of the

edge colours, and requires a knowledge of the associated doubly even code. In cases where

the associated code is unknown, Lemma 8.2.4 suggests an efficient method for finding the

code, and hence relating a given (N, k) Adinkra to its ‘parent’ N -cube Adinkra, in the

sense of Theorem 7.3.7. In particular, Lemma 8.2.4 implies the following result.

172


Corollary 8.3.3. Given an (N, k) Adinkra with related code C, the codewords of C corre-

spond exactly to the cycles of G in which at least 1 edge dimension appears once (modulo

2).

We provide an example of the above certificates below. Consider the two Adinkras of

Fig. 8.2. These are both height-3, (5, 1) Adinkras. They are in the same equivalence

class, however by calculating the above certificate for each we show that they are non-

isomorphic. To analyse the above Adinkras, we first order the edge colours from green to

Figure 8.2: Two equivalent, non-isomorphic (5, 1) Adinkras. They differ precisely in theswitching state of the yellow edges.

purple, such that in the bit-string representation, green corresponds to the first bit, and

purple to the fifth. Alternatively, the green edges are associated with the first Clifford

generator, and the purple edges to the fifth generator. As these are (5, 1) Adinkras, they

have an associated (5, 1) double even code. By inspection, or by explicitly finding the non-

trivial cycles in the Adinkras, we note that the associated code is (11110)†, corresponding

to a 4-length cycle comprising edge colours red, yellow, blue, purple. By Theorem 8.1.5,

†Note that ‘first’ bit of the bit-string is the right-most bit.

173


the induced 1-level Adinkra has two orbits, corresponding to the sets of nodes with the

same parity (inner product modulo 2) relative to this code. Denote the two Adinkras by

G and H respectively. Then by applying the results of Section 8.1, we obtain µ values of:

µG(3, 1) = 2 µH(3, 1) = 0

µG(3, 2) = 0 µH(3, 2) = 2

µG(2, 1) = µG(2, 2) = 4 µH(2, 1) = µH(2, 2) = 4 (8.18)

µG(1, 1) = 2 µH(1, 1) = 4

µG(1, 2) = 4 µH(1, 2) = 2.

In other words, the two height-3 vertices of G are in a different orbit to those of H. Hence

certG 6= certH , and the two Adinkras are in different isomorphism classes.

8.3.1 Partitioning into Isomorphism Classes: Further Examples

In [151] it was shown that simply recording the number of vertices at each height is

not sufficient to characterise Adinkras. In particular, they provide examples of pairs of

Adinkras which are in the same equivalence class, but not isomorphic, despite having the

same number of vertices at each height. In this section, we analyse the examples of [151]

using the techniques described above.

Firstly however, we note that a different definition of isomorphism is considered in [151].

Specifically, they consider the situation where permutations of the edge colours preserve

isomorphism, a variation discussed in Section 7.3.1. Allowing this more general definition

of isomorphism results in several changes to the results of the preceding sections. In

particular, note that the automorphism group of theN -cube Adinkra would simply become

that of the N -cube: the hyperoctahedral group of order 2NN !. A permutation of the

edge colours corresponds to a reordering of the bit-string associated with each vertex (or

equivalently a permutation of the Clifford generators associated each edge dimension).

Hence the automorphism group of (N, k) Adinkras will be extended according to the

following lemma.

Lemma 8.3.4. Allowing permutations of the edge colours to preserve isomorphism, an

(N, k) Adinkra G with associated doubly even code C has an automorphism group (ignoring

height assignments) of order2N

22k+a|Aut(C)|,174


where Aut(C) is the automorphism group of the code C, and where a = 1 if C contains

the all-1 codeword, and a = 0 otherwise.

In other words, any permutations that correspond to symmetries of the code will extend

naturally to automorphisms of the Adinkra. Conversely, if a permutation of the bit-string

is not in the automorphism group of the code, then trivially it cannot be an automorphism

of the Adinkra. Note that Lemma 8.3.4 applies trivially to N -cube Adinkras, for which

k = 0 and C = (00 . . . 0), hence |Aut(C)| = N !.

Hence, Klein flip degeneracy no longer necessarily exhibits for Adinkras containing the

all-1 codeword. Instead, note that any automorphism of the associated doubly even code

corresponding to an odd permutation of the edge-colours (or alternatively the underlying

supercharges) effectively performs a Klein flip, substituting bosons or fermions and vice-

versa. Hence, Klein flip degeneracy occurs if and only if the associated code contains the

all-1 codeword and the automorphism group of the code contains no odd-permutations,

hence narrowing down the possible candidates.

Somewhat fortuitously, a recent result regarding doubly even codes sheds some light on

this situation. Gunther et al. [152] prove that the automorphism group of self-dual doubly

even codes is always contained in the alternating group. Given a doubly even binary (N, k)

code C, its dual C⊥ is given by

C⊥ = v = (v1, . . . , vn) ∈ Zn2 : 〈v, c〉 = 0 ∀ c ∈ C. (8.19)

A self-dual code is simply a code equal to its own dual; a doubly even (N, k) code is self

dual if and only if k = N/2, implying in turn that N = 0 (mod 8). Hence the following

result holds.

Lemma 8.3.5. An (N, k) Adinkra exhibits Klein flip degeneracy† if N = 2k and N = 0

(mod 8).

However, the question of whether any other Adinkras with Klein flip degeneracy† exist

still remains. This is equivalent to asking whether any doubly even codes exist which are

both not self-dual and contain no non-trivial even-permutation automorphisms, and is the

subject of continuing work. Furthermore, if the automorphism group of doubly even codes

turns out to be sufficiently rich, a greatly simplified certificate encoding the isomorphism

class of an Adinkra could be developed, as will be discussed briefly in Section 8.5.

†Relative to this definition of isomorphism, allowing edge-colour permutations.

175


In [151], two pairs of equivalent but non-isomorphic Adinkras are presented, each pair

having having the same number of vertices at each height. The first pair comprises two

height 3, (5, 1) Adinkras, isomorphic (up to a permutation of edge colours) to those of

Fig. 8.3 below. Note that the second Adinkra of Fig. 8.3 is identical to the first Adinkra

of Fig. 8.2.

Figure 8.3: Two non-isomorphic height 3, (5,1) Adinkras belonging to the same equiva-lence class.

Applying the methods described in the previous section to this pair proceeds as in the

analysis of the Adinkras of Fig. 8.2. As in that example, after ordering the edge colours

from blue to green, each Adinkra has associated code (11110). We see that the top two

nodes of the first Adinkra belong to different orbits of the corresponding 1-level Adinkra,

whereas the top two nodes of the second Adinkra belong to the same orbit. Hence the

two Adinkras are trivially distinguished. In particular, following the conventions in the

equations of Example 8.18, the µ values of these two Adinkras, labelled by G and H

176


respectively, are:

µG(3, 1) = 1 µH(3, 1) = 2

µG(3, 2) = 1 µH(3, 2) = 0

µG(2, 1) = µG(2, 2) = 4 µH(2, 1) = µH(2, 2) = 4 (8.20)

µG(1, 1) = 3 µH(1, 1) = 2

µG(1, 2) = 3 µH(1, 2) = 4.

Hence certG 6= certH , and the two Adinkras are in different isomorphism classes. Also,

note that the ‘orbit spread’, the number of vertices in each orbit at each height, has a

different character in each Adinkra. So regardless of the ordering of the orbits, these two

Adinkras must remain in different isomorphism classes if edge-colour permutations are

allowed.

The second pair of Adinkras in [151] comprises two height 3, (6, 2) Adinkras, isomorphic

to the pair displayed in Fig. 8.4 below.

Figure 8.4: Two non-isomorphic height 3, (6,2) Adinkras belonging to the same equiva-lence class.

177

8.4. NUMERICAL RESULTS

In this case, after ordering the edge colours from dark blue to green, as for the top left node

of the first Adinkra, both Adinkras have the associated code generated by

(1 1 1 1 0 0

0 0 1 1 1 1

),

and hence each have four orbits in the automorphism group of the corresponding 1-level

Adinkras. The top two nodes of the first Adinkra are connected via the length-2 paths

(001001) and (000110), having odd inner product with each of the codewords in the gen-

erating set above. Hence these nodes are in different orbits of the automorphism group.

Conversely, the top two nodes of the second Adinkra are connected via the length-2 paths

(110000), (001100) and (000011), having even inner product with each of the codewords.

Hence they are in the same orbit. As a result, the certificates of Theorem 8.3.2 are dif-

ferent for each Adinkra, hence they are in different isomorphism classes. Again, following

the conventions in the equations of Example 8.18, we obtain µ values of

µG(3, 1) = µG(3, 2) = 1 µH(3, 2) = 2

µG(2, i) = 2, ∀ i ∈ [4] µH(2, i) = 2, ∀ i ∈ [4] (8.21)

µG(1, 1) = µG(1, 2) = 1 µH(1, 2) = 0

µG(1, 3) = µG(1, 4) = 2 µH(1, 1) = µG(1, 3) = µG(1, 4) = 2.

Hence, as in the previous example, certG 6= certH , and moreover the Adinkras remain in

different isomorphism classes if edge-colour permutations are allowed.

8.4 Numerical Results

The automorphism group results of the preceding sections were also verified numerically,

independently to the analytical results. All (N, k) Adinkras up to N ≤ 16 were produced,

and the related automorphism group and equivalence classes were calculated for each such

Adinkra. The results found were consistent with the analytical results described in the

earlier sections. In particular:

• All (N, k) Adinkras with the same associated code C were found to be in the same

equivalence class, except in the cases where (11 . . . 1) ∈ C. In these cases, the

Adinkras were split into two equivalence classes, related via the Klein flip operation,

as described in Corollary 8.1.8.

178


• All 1-level (N, k) Adinkra had 2k orbits, each consisting of sets of vertices with the

same set of inner products with the codewords in C, according to Theorem 8.1.5.

All doubly even (N, k) codes up to N = 28 (and many larger parameter sets) can be

found online (Miller, [153]). For each parameter set (N, k), the two possibly inequivalent

standard form Adinkras corresponding to the construction method of Theorem 7.3.11 were

produced relative to each doubly even code on these parameters. As we were considering

only equivalence classes and automorphism groups of valise Adinkras, it sufficed to use the

adjacency matrix form defined in Section 8.2 for this analysis, as height assignments need

not be encoded in the representation. The orbits were then calculated from the adjacency

matrices by forming a canonical form relative to each node via a set of switching operations

and a permutation of the vertex set. A canonical form is defined to be a mapping π,

consisting of a permutation of the vertex labels and set of vertex switching operations,

such that for any two Adinkras G and H, π(G) = π(H) if and only if G and H are

isomorphic.

Then given a vertex v and adjacency matrix A of an (N, k) Adinkra (together with an

ordering of the edge colours), we define the following canonical form of A relative to v.

(i) Permute the ordering of the vertices relative to their connections to v and the or-

dering of edge colours, such that the vertices are ordered:

(v, i1v, . . . , iNv, i2i1v, . . . , iN i1v, i3i2v, . . . , iN iN−1 . . . i1v).

(ii) Switch vertices (i1v, . . . , iNv) such that all edges of v have even parity.

(iii) Repeat for neighbours of vertex i1v.

(iv) Repeat for each vertex, in the ordering above, such that edges appearing lexico-

graphically earlier in the adjacency matrix are of even parity where possible.

This defines a unique switching state of the Adinkra, up to isomorphism. Furthermore it

is a canonical form, in that any two vertices belonging to the same orbit result in identical

matrix forms. To illustrate the above process, we consider a (6, 2) Adinkra with associated

code generated by

(1 0 1 1 1 0

0 1 1 1 0 1

). The standard form valise Adinkra is shown below, where

the top leftmost vertex has bit-string (0000), and its edges are ordered lexicographically

from left to right.

179

8.4. NUMERICAL RESULTS

Consider the process of converting this Adinkra to canonical form, relative to vertex (0011)

(the fifth white vertex from the left in the above figure). If we order the vertices by height

first, then left to right, step (i) corresponds to the set of vertex lowering operations leading

to the Adinkra:

Steps (ii) and (iii) correspond to a set of switching operations, permuting the switching

state in the following stages:

=⇒

Step (iv) then involves the remaining set of transformations:

=⇒

At which point the Adinkra is now in canonical form. The particular switching state of

this resulting Adinkra is unique to choices of source vertices in the same orbit, in this case

the set of vertices (0000), (0011), (1110), (1101).

180


8.5 Conclusions and Future Work

This work provides a graph theoretic characterisation of Adinkras, in particular classifying

their automorphism groups according to an efficiently computable set of local parameters.

In the current work, a number of comparisons are made to previous work. The connection

between Adinkras and codes [30] has been re-examined and found to be robust. However,

this connection is also exploited to utilise the standard form leading to more computational

efficient algorithms for the study of Adinkras. As well, the observations based on matrix

methods used within the context of d = N = 4 [32] have now been extended by a formal

proof to all values of d and N . Also as emphasised in Section 8.4, numerical studies up to

values of N = 16 provide additional concurrence. These results support the proposal that

χ0 ‘chi-null’ is a class valued function defined on valise Adinkras.

All non-valise Adinkras, through a series of node raising and lowering can be brought

to the form of a valise Adinkra. In this sense χ0 is defined for all Adinkras. However,

for non-valise Adinkra, χ0 is not sufficient to define classes. For this purpose, the new

certificate µA(h, a), where A is an arbitrary Adinkra, seems to fill in a missing gap.

Additionally, the classification method used here is robust in the sense that it can be

readily adapted to address a variety of definitions of isomorphism/equivalence, relevant to

the particular supersymmetric system of interest. For instance, in Section 8.3.1 the gen-

eralised definition of isomorphism including edge-colour permutations yields a generalised

certificate.

8.5.1 Open Questions

It is the work of future investigations to explore whether these tools (χ0 and µA(h, a)) are

sufficient to attack the problem of the complete classification of one-dimensional off-shell

supersymmetrical systems. One obvious future avenue of study is to investigate the role

codes play in Gnomon Adinkras [150] and Escheric Adinkras [5]. This as part of continuing

to attack the general problem presents continuing challenges.

Note that when edge-colour permutations are allowed, the calculation of the certificate

encoding the isomorphism class of an Adinkra, as detailed in Section 8.3 is currently

more complex, compared to the situation when edge-colour permutations do not preserve

isomorphism, in that the automorphism group of the doubly even code must also be cal-

culated. One future direction of research is the investigation of the properties of these

181


automorphism groups. It is already known that self-dual doubly even codes have auto-

morphism group contained in the alternating group [152], however further properties must

be discovered to determine all possible Klein flip degenerate Adinkras in this case.

Specifically, if it can be shown that all doubly even codes which are not self dual have non-

trivial even-permutation automorphisms, or more specifically that this at least holds for

doubly even codes containing the all-1 codeword, then Adinkras with Klein flip degeneracy

will have been classified. This is part of ongoing work; a cursory search of a database of

doubly even codes up to N = 32 [153] has revealed no counterexamples as of yet. To this

point, very little is known regarding automorphism group properties of general doubly even

codes, with the work of [152] mentioned in Section 8.3.1 being one of the few exceptions.

Whilst it is unknown whether or not non-self-dual doubly even codes must have even

automorphisms, sporadic cases have been found of extremal self-dual codes with trivial

automorphism groups, most notably in the work of [154].

If the automorphism groups of general non-self-dual doubly even codes are known to

be sufficiently rich enough, the certificate described in Section 8.3.1 could potentially be

significantly simplified, to the point where it could perhaps be expressed independently to

the automorphism group of the code.

182

Chapter 9

Conclusion

This thesis explores a range of questions relating to classification of graphs and quantum

walk based search problems. The efficiency of both quantum circuit implementations

of quantum walks and quantum walk based search algorithms is related to the level of

symmetry in the underlying graph. Several families of graphs are presented that are

shown to be amenable to efficient quantum walk based searching. For each such family,

explicit quantum circuit implementations are constructed. The case of quantum search

for marked subgraphs of a hypercube is looked at in depth. In this case I show that

marked sub-cubes of dimension d within an n-cube can be efficiently found in the case

where n−d is sufficiently large, generalising the results of Shenvi et al. [2]. Quantum walk

based searching along a family of twisted toroids is also considered. In this case numerical

arguments are provided, showing that the quantum walk algorithm finds marked nodes

in O(√n) time, along a graph of order n. It is the aim of future work to extend these

numerical results to an analytical proof, as done in the hypercube case.

Separately, the level of structural information provided by walking on a graph is explored,

leading to the construction of a novel quantum walk based graph isomorphism algorithm.

This is shown to efficiently characterise all strongly regular graphs up to order 64 as well

as a range of other graphs tested, including vertex-transitive graphs and projective planes.

The quantum algorithm employs a novel measurement method to directly compare the

amplitude distributions related to each graph (of order n) using a single measurement.

Any differences of sufficiently large amplitude† existing between these distributions can

be distinguished up to an arbitrarily high probability with this single measurement, via

†Here ‘sufficiently large’ implies that the differences are not exponentially small. In other words, forgraphs on n vertices, they scale with O(nc), for some constant c.

183

extending the related quantum walk by O(log(n)) steps. I relate this algorithm to a well-

known classical algorithm, demonstrating that in its present form it does not suffice to

characterise general graphs. The question of whether such a method suffices to distinguish

certain subclasses of graphs, in particular strongly regular graphs, remains open.

After relating this quantum walk based graph identification algorithm to the classical

Weisfeiler-Leman (WL) method, this classical approach is explored in depth. Although

the WL method initially appeared to be a promising approach towards solving the graph

isomorphism (GI) problem, it was shown insufficient to classify general graphs in Cai, Furer

and Immerman [3] and Evdokimov and Ponomarenko [4], in which families of graphs of

order n were constructed requiring O(exp(n)) time to distinguish.

Whilst the counterexample graph pairs of [3,4] cannot be distinguished by the k-dim WL

method, I consider the case of characterising individual graphs using the recursive k-dim

WL method. I prove that the graphs CFI(G) and X(G) (the counterexample graphs

constructed by [3] and [4] respectively) will be individually characterised by the recur-

sive (k + 1)-dim WL method, provided that the original graph G is characterised by the

recursive k-dim WL method†.

Hence the direct graph types constructed in [3,4] do not necessarily provide counterexam-

ples to the recursive WL method. Of course directly addressing the recursive k-dim WL

method was not the purpose of either of these works, and by itself this does not constitute

a significant result, in that trivial extensions of these graphs can be constructed which

provably do constitute counterexamples‡.

However, this result does become significant when combined with the decomposition

method of Section 6.4, in which it is proven that such composite graphs that are also

counterexamples to the recursive k-dim WL method and additionally satisfy the assump-

tions of Section 6.4.4, will be characterised by an extended k-dim WL method which

includes this decomposition process.

Part of the significance of these results is that there are no longer any known counterex-

amples to this extended WL method. The constraints imposed on the initial composite

graphs to facilitate the proofs of Section 6.4 do not seem particularly onerous, in the sense

that it does not appear easy to circumvent them, finding graphs for which they are not

†In fact it is additionally shown that the graph X(G) will trivially always be characterised by therecursive k-dim WL method, due to the method of colouring these graphs.‡For instance, the join of CFI(G) and CFI

′(G) will trivially not be distinguished by the recursive k-dim

WL method in the case where CFI(G) is k-similar - i.e. when G has no separator of size k.

184

satisfied. It is the focus of future efforts to investigate general properties of the ‘coun-

terexample’ k-equivalent graphs, together with attempting to impose further constraints

on graphs for which the decomposition method of Section 6.4 does not apply. Finding

such graphs would represent a true advance in the study of the WL method, as they must

possess novel properties, presumably intimately related to the property of k-equivalence.

Subsequent to the classical WL method work, I present a further implementation of gen-

eral graph characterisation methods, efficiently classifying the automorphism groups of a

family of graphs known as Adinkras, introduced by Faux and Gates [5] to study off-shell

representations of supersymmetry. A set of local parameters is shown to classify Adinkras

according to their equivalence and isomorphism classes, representing further progress in

this recent pursuit of a solution to the long-standing ‘off-shell’ problem of supersymmetry.

Previous results dealing with characterisation of Adinkra degeneracy via matrix prod-

ucts are extended, and algorithms presented for calculating the automorphism groups of

Adinkras and partitioning Adinkras into their isomorphism classes.

These results regarding Adinkras leave several unanswered questions, which are the focus

of ongoing work. The properties of Adinkras have been linked to that of a corresponding

doubly even code in the work of Doran et al. [26, 27]. However several highly relevant

properties of these codes themselves remain unknown. In particular, a solution to questions

regarding the automorphism group properties of general doubly even codes may allow for

a further simplification to the characterisation methods described here.

185

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