Wei Wang Xi’an Jiaotong University

Generalized Spectral Characterization of Graphs:

Revisited

Shanghai Conference on Algebraic Combinatorics (SCAC), Shanghai, Aug, 2012

Outline

IntroductionReview of Some Old Results Some New ResultsSummaryAn Open Problem for Further Research

IntroductionThe spectrum of a graph encodes a lot of

information about the given graph, e.g., From the adjacency spectrum, one can deduce (i) the number of vertices, the number of edges; (ii) the number of triangles ; (iii) the number of closed walks of any fixed length; (iv) bipartiteness; …………… From the The Laplacian spectrum, one can deduce : (i) the number of spanning trees; (ii) the number of connected components; …………….Question: Can graphs be determined by the

spectrum?

Cospectral GraphsA pair of cospectral graphs;

Schwenk (1973): Almost no trees are determined by the spectrum.

DS GraphsQuestion: Which graphs are determined by

their spectrum (DS for short)?This is an old unsolved problem in Spectral

Graph Theory that dates back to more than 50 years .

Applications: Chemistry; Graph Isomorphism Problem; The shape and sound of a drum (“Can one

hear the shape of the drum?”); ……

Two recent survey papers : E. R. van Dam, W. H. Haemers, Which graphs are

determined by their spectrum? Linear Algebra Appl. 373 (2003) 241-272.

E. R. van Dam, W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Mathematics, 309 (2009) 576-586.

This talk will focus on the topic of characterizing graphs by both the spectrum and the spectrum of the complemnt of a graph.

Notations and Terminologies : a simple graph (unless stated otherwise) vertex set ; edge set

.The adjacency matrix of graph G is an

matrix with .

The characteristic polynomial of graph G: The spectrum of G is the multiset of all the

eigenvalues of Two graphs are cospectral if . denotes a prime number, Fp denotes a finite field

with p elements, denotes the rank of W over Fp.

),( EVG

)()( ijaGA nn

jiij vandvbetweenedgesofnumberthea

},...,,{ 21 nvvvV },...,,{ 21 meeeE

))(det();( GAIG

)(GA

HG, );();( HG

p

)(Wrank p

DGS GraphsTwo graphs are cospectral w.r.t. the generalized

spectrum if and .

A graph is said to be determined by the generalized spectrum (DGS for short), if any graph that is cospectral

with G w.r.t. the generalized spectrum is isomorphic to .

G

HG,

);();( HG );();( HG

G

G

DGS Graphs: An Review of Some Old ResultsThe walk-matrix of graph G:

where is the all-one vector.Remarks: 1. The -th entry of W is the number of

all walks starting from vertex with length . 2. The arithmetic properties of det(W) is

crucial for our discussions.

])(,...,)(,[ 1eGAeGAeW nTe )1,...,1,1(

),( ji

i 1j

Controllable Graph

A graph G is called a controllable graph if the corresponding walk-matrix is non-singular.

The set of all controllable graphs of order is denoted by

W

n

A Simple Characterization

Theorem 1. [Wang and Xu ,2006] Let . Then there exists a graph H with and if and only if there exists a unique rational orthogonal matrix Q such that

(1)

eQeHAQGAQT ),()(

);();( HG );();( HG

A Simple CharacterizationDefine

.

Theorem 2. [Wang and Xu, 2006] Let . Then G is DGS if and only if contains only permutation matrices.

Question: How to find out all ?

The Level of QDefinition: Let Q be a rational orthogonal

matrix with Qe=e, the level of Q is the smallest positive integer such that is an integral matrix.

If , then Q is a permutation matrix.

Example:

The Smith Normal FormAn integral matrix is called unimodular if

.Let be an n by n integral matrix with full

rank. Then there exist two unimodular matrices and such that

where is called the i-th elenmentry divisor of .

V

d

d

d

UM

n

2

1

1)det( U

.1,..,2,1,| 1 nidd ii

id

U

U V

M

M

An Exclusion PrincipleLemma . [Wang and Xu ,2006] Let W be

the walk-matrix of a graph .Let Then we have

Some Basic Ideas

(i) All the possible prime divisors of is finite; they are the divisors of , and hence are divisors of .

(ii) Some of the prime divisors of may not be divisors of , they can be excluded from further consideration.

(iii) If all the prime factors of are not divisors of ,,

then we must have =1, and hence contains only permutation matrices and G is DGS.

nd )det(W

)det(W

)det(W

Primes p>2Let be a prime, . If Eq. (1) has

no solution, then is not a divisor of .Assume , the solution to the

system of linear equations can be written as over finite field Fp .

If over Fp , then is not a divisor of .

Using this way, the odd prime divisors of can be excluded in most cases.

2p ndp |

1)( nWrank p

)0( kx

0 T p

p

nd

Examples

The First GraphIt can be computed .For p=17,67,8054231, solve Eq (1) and

check whether is zero or not over Fp , .All primes (except p=2) can be excluded.

1492735 ,25 ,12 T

T

805423167172)( 112 Gd

The Second GraphIt can be computed .For p=3,5,197,263,5821, solve Eq (1) and

check whether is zero or not over Fp , .All primes (except p=2 ,5) can be excluded.

4298 ,101 ,139 ,0 ,1 T

T

5821263197532)( 2213 Gd

The prime p=2When p=2, however, the system of linear

equations has always non-trivial solutions. Thus, p=2 cannot be excluded using above method.

To exclude p=2, we have to develop more intensive exclusion conditions. I shall not go into the details.

To conclude, it can be shown that the first graph is DGS. But cannot be shown to be DGS, since p=5 cannot be excluded by using the existing methods.

1G

2G

Question: Does there exist a simple method to exclude the primes p>2? (The case p=2 is more involved, I shall concentrate on the case p>2 in this talk.)

A New Exclusion Principle for p>2

Theorem 3. [Wang,2012] Let . Let Suppose that , where

is an odd prime. Then is not a divisor of .

2p

p

eQeHAQGAQT ),()()det(|| Wp

)(mod pp

zWxW

TT )det(|2 Wp

);();( HG

);();( HG

A diagram for the Proof of Theorem 3

The Proof: A sketch

Example and counterexampleIn previous example, is DGS, since p=5

can be excluded by Theorem 3.Theorem 3 may be false if the exponents of

p>2 is larger than 1.Let the adjacency matrix of G be given as

follows, is a (0,1)matrix, and hence is an adjacency

matrix of another graph H. However, note that

Thus, p=3 cannot be excluded.

AQQT

2G

An Counter-example

.

Summary

We have reviewed some existing methods for showing a graph to be DGS; in particular, we review the exclusion principle for excluding those odd primes of det(W).

We also present a simple new criterion to exclude all odd prime factors with exponents one in the prime decomposition of det(W);

It suggests that the arithmetic properties of det(W) contains much information about whether G is DGS or not.

Problem for Further Research

Conjecture [Wang,2006]. Let . Then G is DGS if

(which is always an integer) is square-free.

Remarks: i) We have shown that odd primes p>2 with

exponents one can be excluded. ii) The case p=2 still needs further

investigations!!!

References [1] E. R. van Dam, W. H. Haemers, Which graphs are

determined by their spectrum? Linear Algebra Appl., 373 (2003) 241-272. [2] E. R. van Dam, W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Mathematics, 309 (2009)

576-586. [3] W. Wang, C. X. Xu, A sufficient condition for a family of graphs being determined by their generalized spectra,

European J. Combin., 27 (2006) 826-840. [4] W. Wang, C.X. Xu, An excluding algorithm for testing whether a family of graphs are determined by their generalized

spectra, Linear Algebra and its Appl., 418 (2006) 62-74. [5] W. Wang, On the Spectral Characterization of Graphs, Phd

Thesis, Xi'an Jiaotong University, 2006. [6] W. Wang, Generalized spectral characterization of graphs,

revisited, manuscript, Aug, 2012.

Thank you!The end!