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GROUPS AND SYMMETRY CHAPTER 2

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GROUPS AND SYMMETRY

CHAPTER 2

Contents

1. Groups and subgroups

2. Group actions

3. Symmetry groups

4. Covering graphs

5. Symmetry in graphs

6. Symmetry in metric spaces

7. Representation of graphs and their symmetries

1. Gropus and Subgroups

Groups

• Group G consists of elements, g1, g2, ... and an operation ². It satisfies the following:

• A1. Replacing any two symbols in the equation a ² b = c by group elements uniquely determines the third one.

• A2. For any three group elements gi,gj,gh we have: (gi ² gj) ² gk = gi ² (gj ² gk)

Usual Group Axioms

• (a ² b) ² c = a ² (b ² c)

• There exists e, such that for any a a ² e = e ² a = a.

• For each a there exists a’ such that a ² a’ = a’ ² a = e.

• Exercise: Show that both systems of axioms are equivalent.

Finite and Infinite Groups

• The number |G| is called the order of group G.

• The groups of finite order are called finite groups. All other groups are called infinite.

Abelian Group

• If a ² b = b ² a for any a,b 2 G the group G is called Abelian or commutative.

Residues mod n: Zn.

• Two views:

• Zn = {0,1,..,n-1}.

• Define ~ on Z:

• x ~ y $ x = y + cn.

• Zn = Z/~.

• (Zn,+) an abelian group, called the cyclic group of order n. Here + is taken mod n!!!

Example (Z6, +).

+ 0 1 2 3 4 5

0 0 1 2 3 4 5

1 1 2 3 4 5 0

2 2 3 4 5 0 1

3 3 4 5 0 1 2

4 4 5 0 1 2 3

5 5 0 1 2 3 4

Example (Z6, £) is not a group.

£ 0 1 2 3 4 5

0 0 0 0 0 0 0

1 0 1 2 3 4 5

2 0 2 4 0 2 4

3 0 3 0 3 0 3

4 0 4 2 0 4 2

5 0 5 4 3 2 4

Example(Z6\{0}, £) is not a group.

£ 1 2 3 4 5

1 1 2 3 4 5

2 2 4 0 2 4

3 3 0 3 0 3

4 4 2 0 4 2

5 5 4 3 2 4

For p prime, (Zp\{0}, £) forms a group.

R - real numbers

• (R,+) is a group• The group is Abelian and infinite. Its unit is

0, the inverse of a is –a.• (R \ {0},.) is a group• The group is infinite, Abelian, its unit is 1,

the inverse of a is 1/a.• Let R+ := {x 2 R| x > 0}. (R+,.) is also a

group.

Subgroup

• H µ G is a subgroup if H is a group for the same group operation.

• There are subsets, closed for the group operation, that are not subgroups. For instance, (N,+) is not a subgroup of (Z,+).

Cosets

• G – group

• H – subgroup

• The set aH := {a ² x| x 2 H} is called (left) coset of H in G.

• G = H t aH t bH t ...

Index

• Let H µ G be a subgroup of G.

• [G:H] := # cosets of H is called the index of H in G. For finite groups [G:H] = |G|/|H|.

Q – rational numbers

• (Q,+) is a group.

• Rational numbers form a group under addition.

• (Q \ {0},.) is a group.

Z - integers

• (Z,+) is a group.

• (Z \ {0},.) is not a group.

Complex numbers C.

• = a + bi 2 C.

• * = a – bi.

• = c + di 2 C.

• = (ac –bd) + (bc + ad)i.

• 0, / = [(ac + bd) + (bc – ad)i]/[c2 + d2].

• -1 = (a –bi)/(a2 + b2).

C, - Complex numbers

• (C \ {0},.) is a group

• (C,+) is a group

Quaternions H.

• Quaternions form a non-commutative field.• General form:• q = x + y i + z j + w k., x,y,z,w 2 R.• i 2 = j 2 = k 2 =-1.

• q = x + y i + z j + w k.• q’ = x’ + y’ i + z’ j + w’ k.

• q + q’ = (x + x’) + (y + y’) i + (z + z’) j + (w + w’) k.• How to define q .q’ ?• i.j = k, j.k = i, k.i = j, j.i = -k, k.j = -i, i.k = -j.• q.q’ = (x + y i + z j + w k)(x’ + y’ i + z’ j + w’ k)

(Q,.) – The Quaternion Units

QQ 11 -1-1 ii -i-i jj -j-j kk -k-k

11 1 -1 i -i j -j k -k

-1-1 -1 1 -i i -j j -k k

ii i -i -1 1 k -k -j j

-i-i -i i 1 -1 -k k j -j

jj j -j k -k -1 1 i -i

-j-j -j j -k k 1 -1 -i i

kk k -k j -j -i i -1 1

-k-k k -k j -j -i i 1 -1

Conjugation

• Given a subgroup H of a group G, then for any g 2 G define: H’ = g-1Hg := {g-1hg|h 2 H}.

• H’ is called a conjugate of H.• H’ is a subgroup of G.• Conjugation is an equivalence relation on

the set of subgroups of G.• |H’| = |H|.

Normal Subgroup

• If H · G has no nontrivial conjugates, it is called normal.

• For a normal group the quotient G/H forms also a group.

• G/H = {Ha|a 2 G}

• |G/H| = [G:H].

Group Homomorphisms and Isomorphisms

• :G1 ! G2 is a group homomorphism if for any g,h 2 G1 we have: (g ² h) = (g) ± (h), where (G1,²) and (G2,±) are groups.

• If, in addition, is bijection, then it is called an isomorphism.

Exercises 1-1

• N1: There is only one way to complete the definition of multiplication of quaternions and respect distributivity!

• N2: Represent quaternions by complex matrices (matrix addition and matrix multiplication)! Hint: q = [ ; -* *].

Exercises 1

• N3. What is the index [R\{0}:R+\{0}] for the multiplicative group.

• N4. Show that the multiplicative group {1,-1,i,-i} µ C is isomorphic to Z4.

• N5. Determine all subgroups of Zn.

2. Group Actions

Symmetric Group Sym(n)

• As we know a permutation is a bijective mapping of a set A onto itself: : A A. Permutations may be multiplied and form the symmetric group Sym(A) = Sym(n) = Sn = SA, that has n! elements, where n = |A|.

Permutation Group

• Any subgroup G · Sym(A) is called a permutation group. If we consider an abstract group G then we say that G acts on A.

• In general the group action is defined as a triple (, A, ), where is a group, A a set and : ! Sym(A) a group homomorphism.

• In general we are only interested in faithful actions, i.e. actions in which is an isomorphism between and ().

Automorphisms of Simple Graphs

• Let X be a simple graph. A permutation h:V(X) ! V(X) is called an automorphism of graph X if for any pair of vertices x,y 2 V(X) x~y if and only if h(x)~h(y). By Aut X we denote the group of automorphisms of X.

• Aut X is a permutation group, since it is a subgroup of Sym(V(X)).

Orbits and Transitive Action

• Let G be a permutation group acting on A and x 2 A. The set [x] := {g(x)|g 2 G} is called the orbit of x. We may also write G[x] = [x].

• G defines a partition of A into orbits: A = [x1] t [x2] t ... t [xk].

• G acts transitively on A if it induces a single orbit.

Example

• Aut G(6,2) induces two orbits on the vertex set.

• Aut G(6,2) also induces an action on the edge set. There we get three orbits.

Orbits

• Let act on space V. On V an equivalence relation ¼ is introduced as follows:

• x ¼ y , 9 2 3: y = (x).

• Equivalence, indeed:» Reflexive

» Symmetric

» Transitive

• [x] ... Equivalence class to with x belongs is called an orbit. (Also denoted by [x].)

Example

• Graph G=(V,E) has four automorphisms.

• V(G) ={1,2,3,4} splits into two orbits [1] = {1,4} and [2] = {2,3}.

• E(G) = {a,b,c,d,e} also splits into two orbits: [a] = {a,b,e,d} and [c] = {c}.

1

3 4

2a

e

c bd

Stabilizers and Orbits

• Let G be a permutation group acting on A and let x 2 A. By G(x) we denote the orbit of x.

• G(x) = {y 2 A| 9 g 2 G 3: g(x) = y}

• Let Gx µ G be the set of group elements, fixing x. Gx is called the stabilizer of x and forms a subgroup of G.

Orbit-Stabilizer Theorem

• Theorem: |G(x)||Gx| = |G|.

• Corollary: If G acts transitively on A then |A| is the index of any stabilizer Gx in G.

Burnside’s Lemma

• Let G be a group acting on A.

• For g 2 G let fix(g) denote the number of fixed points of permutation g.

• Let N be the number of orbits of G on A.

• Then:

Regular Actions

• The transitive action of G on A is called regular, if |G| = |A|, or equivalently, if each stabilizier is trivial.

• An important and interesting question can be asked for any transtive action of G on A.

• Does G have a subgroup H acting regularly on A?

Semiregular Action

• Definition: Grup acts on V semiregulary,

• If there exists 2 3: = ( ...) ( ...) ...( ...) composed of cycles of the same size r; |V| = r s.

• For each x 2 V we have: |[x]| = r.

Primitive Groups

• A transitive action of G on X is called imprimitive, if X can be partitioned into k (1 < k < |X|) sets: X = X1 t X2 ,t ... t Xk (called blocks of imprimitivity) and each g 2 G induces a set-wise permutation of the Xi’s.

• If a group is not imprimitive, it is called primitive.

Example

• For a prism graph n, Aut n is imprimitive if and only if n 4.

• There are n blocks of imprimitivity of size 2, each corresponding to two endpoints of a side edge.

Permutation Matrices

• Each permutation 2 Sym(n) gives rise to a permutation matrix P() = [pij] with pij = 1 if j = (i) and pij = 0 otherwise.

• Example: 1 = [2,3,4,5,1] and P(1) is shown below: 0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

1 0 0 0 0

Matrix Representation

• A permutation group G can be represented by permutation matrices. There is an isomorphism P(). And corresponds to P()P(). Since each permutation matrix is orthogonal, we have P(-1) = Pt().

Alternating Group Alt(n)

• A transposition is a permutation interchanging a single pair of elements.

• Permutation is even if it can be written as a product of an even number of transpositions (otherwise it is odd.)

• Even permutations from Sym(n) form the alternating group Alt(n), a subgroup of index 2.

Homewrok 2

• H1. Let X be any of the three graphs below.• Determine the (abstract) group of automorphisms

Aut X.

• Action of Aut X on V(X).

• Action of Aut X on E(X).

X1X2

X3

3. Symmetry Groups

Iso(M)

• Isometries of a metric space (M,d) onto itslef form a group of isometries that we denote by Iso(M).

Sim1(M)

• Similarities of type I of a metric space (M,d) onto itslef form a group of similarities that we denote by Sim1(M).

Sim2(M)

• Type II similarities of a metric space (M,d) onto itslef form a group of similarities that we denote by Sim2(M).

• In any metric space the groups are related:

• Iso(M) · Sim2(M) · Sim1(M).

Symmetry

• Let X µ M be a set in a metric space (M,d). An isometry 2 Iso(M) that fixes X set-wise: (X) = X, is called a (metric) symmetry of X.

• All symmetries of X form a group that we denote by IsoM(X) or just I(X). It is called the symmetry group of X.

• Note: this idea can be generalized to other groups and to other structures!

Free Group F()

• Let be a finite non-empty set. Form two copies of it, call the first +, and the second -. Take all words (+ t -)* over the alphabet + t -. Introduce an equivalence relation in such a way that two words u v if and only if one can be obtained from the other one by a finite series of deletion or insertion of adjacent a+a- or a-a+.

• Let F() = (+ t -)* / . Then F() is a group, called the free group generated by .

• We also denote F() = < | >.

Finitely Presented Groups

• Let and < | > be as before. Let R = {R1, R2, ..., Rk} ½ (+ t -)* be a set of relators.

• The expression < | R> is called a group presentation. It defines a quotient group of < | >.

• Two group elements from F() are equivalent if one can be obtained from the other by successive insertion or deletion of relators or their inverses.

• Since both sets and R are finite, the group is finitely presented.

Generators

• Let G be a group and X ½ G. Assume that X = X- and 1 X. Then X is called the set of generators. Let <X> denote the smallest subgroup of G that contains X. We say that X generates <X>.

Cayley’s Theorem

• Theorem. Every group G is isomorphic to some permutation group.

• Proof. For g 2 G define its right action on G by x xg. The mapping from G to Sym(G) defind by g (x xg) is an isomorphism to its image.

Cyclic Group Cyc(n)

• Let G = <a| an>. Hence G = {1,a,a2,..,an-1}. By Calyey’s Theorem we may represent a as the cyclic permutation (2,3,...,n,1) that generates the group Cyc(n) · Sym(n).

• Note that Cyc(n) is isomorphic to (Zn,+).

• Cyc(n) may also be considered as a symmetry group of some polygons. Cyc(8) is the symmetry group of the polygon on the left.

Dihedral Group Dih(n)

• Dihedral group Dih(n) of order 2n is isomoprihc to the symmetry group of a regular n-gon.

• For instance, for n=6 we can generate it by two permutations: = (2,3,4,5,6,1) and = (1,2)(3,6)(4,5). Dih(n) has the following presentation:

• <s,t|sn=t2=stst=1>

54

3

2 1

6

Symmetry of Platnoic Solids

• There are five Platonic solids: Tetrahedron T, Octahedron O, Hexaedron H, Dodecahedron D and Icosahedron I.

Tetrahedron

• A tetrahedron has • v = 4 vertices, • e = 6 edges and • f = 4 faces.• Determine its

symmetry group.

Octahedron

• An octahedron has • v = 6 vertices, • e = 12 edges and • f = 8 faces.• Determine its

symmetry group.

Hexahedron

• A cube has• v = 8 vertices • e = 12 edges and • f = 6 faces.• Determine its

symmetry group.

Dodecahedron

• A dodecahedron has • v = 20 vertices, • e = 30 edges and • f = 12 faces.• Determine its

symmetry group.

Icosahedron

• An icosahedron has • v = 12 vertices, • e = 30 edges and • f = 20 faces.• Determine its

symmetry group.

Skeleton of Tetrahedron – TS = K4

• K4 has

• v = 4 vertices, • e = 6 edges • f = 4 triangles.

• Aut(K4) = S4.

Skeleton of Octahedron – OS = K2,2,2

• OS has • v = 6 vertices, • e = 12 edges

Skeleton of Hexahedron HS =K2 ¤ K2 ¤ K2

• HS ima • v = 8 vertices • e = 12 edges

Skeleton of Dodecahedron DS = G(10,2)

• G(10,2) has• v = 20 vertices, • e = 30 edges

Skeleton of Icosahedron IS

• Is has • v = 12 vertices, • e = 30 edges

Platonic Solids and Symmetry

• We only considered the groups of direct symmetries (orientation preserving isometries).

• The full group of isometries coincides (in this case) with the group of automorphisms of the corresponding graphs.

• In general: • Sym+(M) · Sym(M) · Aut(MS).

The Escher Problem

Frieze Groups

• Frieze = embroidery from Frieze, horizontal ornamental band (architecture).

• We are interested in symmetry groups of such bands. There are 7 frieze groups.

• We start with a rectangular stamp.

Transformations

• Translation

• Halfturn

• Vertical Reflection

• Glide-reflection

Seven Frieze Types

• Groups (notation):• 11 (translations only)• 12 (translations and halfturns)• m1 (translations and vertical reflections)• 1g (translations and glidereflections)• mg (translations, halfturns, vertical reflections and

glide reflections)• 1m (translations and horizontal reflections)• mm (translations, halfturns,vertical reflections,

glide reflections, horizontal reflections)

11

12

m1

1g

mg

1m

mm

The Groups

• Group Elements• Identity I

• Translation T

• Halfturn R

• Glidereflection G

• Vertical mirror V

• Horizontal mirror H.

• Some relations: R2 = V2 = H2 = I, RV = VR = H,..

Subgroups

• (1) T P C1

• (2) G P,R(P) C1

• (3) T,R B C1 £ D1

• (4) T,V A D1

• (5) T,H S D1

• (6) G,V A,R(A) D1

• (7) all(T,G,R,V,H) H D1 £ D1

Discrete Isometries

• Each metric space M determines the group of distance preserving maps, isometries Iso(M).

• A subgroup of Iso(M) is discrete, if any isometry in it either fixes an element of M or moves it far enough.

• Discrete subgroups of I(R2) fall into three classes:• 17 crystallographic groups

• 7 frieze groups

• finite groups (grups of rosettes).

Theorem of Leonardo da Vinci

• The only finite groups of isometries in the plane are the groups of rosettes (cyclic groups Cn and dihedral groups Dn).

The Escher problem

• There is a square stamp with asymmetric motif.• By 90 degree rotations we obtain 4 different aspects.• By combining 4 aspects in a square 2 x 2 block, a

translational unit is obtained that is used for plane tiling. Such a tiling is called a pattern.

• Question: What is the number of different patterns ?• Answer: 23.

Example

Recall Burnside’s Lemma.

• Let G be a group, acting on space S.

• For g 2 G let fix(g) denote the number of points from S fixed by g.

• Let N denote the number of orbits of G on S.

• Then:

Application

• Determine the group (G) and the sapce (S).

• Pattern can be translated and rotated.

• Basic observation: • Instead of pattern consider the block (signature).

• Group operations:• H – horizontal translation

• V – vertical translation

• R – 90 degrees rotation.

(Abstract) group G

• h2 = v2 = r4 = 1.• hv = vh• hr = rv

v vr vr2 vr3

1 r r2 r3

h hr hr2 hr3

hv hvr hvr2 hvr3

Space S

• Space S consists of 4 4 4 4 = 256 signatures.

• Count fix(g) for g 2 G.

• For instance:• fix(1) = 256.

• fix(r) = fix(r3) = 4.

• fix(h) = fix(v) = 16.

• By Burnside’s Lemma we obtain N = 23.

1-dimensional Escher problem

• Rectangular asymmetric motiff

• Only two aspects.

• 1 x n block (signature)

• Determine the number of patterns:

• Two more variations:• II two motiffs(mirror

images)

• III reflections are allowed.

Space S

• Space S consists of 2 2 .... 2 = 2n signatures.

• Count fix(g) for g 2 G.

• For instance:• fix(1) = 256.

• fix(r) = fix(r3) = 4.

• fix(h) = fix(v) = 16.

• By Burnside’s Lemma we obtain N = 23.

Solution for the basic case

• where g(n) = 0 for odd n and for even n:

nk

kn ngkn

nf|

/ ))(2)((2

1)(

2/2)2/()( nnng

Program in Mathematica

• f[n_] := (Apply[Plus,Map[EulerPhi[#] 2^(n/#)&,Divisors[n]]] + If[OddQ[n],0,(n/2) 2^(n/2)])/(2n)

• f[n_,m_] := (Apply[Plus,Map[EulerPhi[#] (2 m)^(n/#)&,Divisors[n]]] + If[OddQ[n],0,(n/2) (2 m)^(n/2)])/(2n)

• g[n_] := (Apply[Plus,Map[If[OddQ[#],1,2] EulerPhi[#] 4^(n/#)&,Divisors[n]]] + If[OddQ[n],0,(n) 4^(n/2)])/(4n)

Results for a tape

• n I II III• 1 1 2 1• 2 2 6 4• 3 2 12 6• 4 4 39 23• 5 4 104 52• 6 9 366 194• 7 10 1172 586• 8 22 4179 2131• 9 30 14572 7286• 10 62 52740 26524

17 CRYSTALLOGRAPHIC GROUPS6-števna os?

zrcaljenje? 4-števna os?

p6mm p6

Zrcala v 4 smereh?

zrcaljenje?

p4

3-števna os?

p4mm p4gm

zrcaljenje?2-števna os?

zrcaljenje?

glide?Rombska mreža?glide?Drugo zrcalo?

Rombska mreža?

zrcaljenje?

p1pgpmcmp2c2ggp2mg

3-osi na zrcalih?

p31mp3m1

p3

c2mm p2mm

p1

• p1 = <a,b|ab=ba>

p2

• p2 = <a,b,c| b2=c2=(ab)2=(ac)2=1>

pm

• pm = <a,b,c| b2=c2=1, ab=ba, ac=ca>

pg

• pg = <a,b|ab=ba-1>

cm

• cm = <a,b,c| b2=c2=1, ab=ca>

p2mm

• p2mm = <a,b,c,d| a2=b2= c2=d2= 1, (ab)2=(ad)2=(bc)2=(cd)2=1>

p2mg

• p2mg = <a,b,c| b2 = c2 = 1, (ab)2=(ac)2=1>

p2gg

• p2gg = <a,b| (ab)2=1>

Exercises 3-1

• N1. Explain how the Frieze groups can be described by the four letters (aspects of the pattern):

• b, p, q, d.

b

p

q

d

Exercises 3-2

• N2. Determine the Cayley graph of each of the Frieze groups.

• N3. Determine the crystallographic groups that may arise from the classical Escher problem.

Homework 3-1

• H1. Determine the group of symmetries of the prism 6.

• H2. Determine the group of symmteries of the antiprism A6.

• H3. Determine the group of symmetries for the pyramid P6.

• H4. Determine the group of symmetries of the double pyramid B6.

• H5. Generalize to other values of n.

• H6. Repeat the problems for the corresponding skeleta.

Homework 3-2

• Consider the Escher problem with the motiff on the left.

• H7. Determine the abstract group and its Cayley graph.

• H8. What is the number of different patterns?

• H9. What is the number of different patterns if reflections are allowed?

• H10. What is the number of different patterns in the original Escher problem if reflections are allowed?

4. Covering Graphs

Covering Graphs

• Motivation:

• Suppose you are taken to two different labyrinths. Is it possible to tell they are distinct just by walking around?

• Let us call the first graph maze X, and the second one Y.

Question

• Is it possible to distinguish between the two mazes?

• Answer: Yes, we can. In the upper maze there are two adjacent trivalent vertices. This is not the case in the lower maze.

Local Isomorphism

• On the other hand we cannot distinguish (locally) between the upper and lower graph.

• To each walk upstairs we can associate a walk downstairs.

One More Example

• C4 over C3 is no good (Why not?). However, C6 over C3 is Ok.

Fibers and Sheets.

• We say that C6 is a two-sheeted cover over C3. Red vertices are in the same fiber. Similarly, the dotted lines belong to the same fiber.

• The graph mapping f: C6 C3 is called covering projection.

• The pre-image of a vertex f-1(v) (or an edge f-1(e)) is called a fiber.

• The cardinality of a fiber is constant. k =|f-1(v)| is called the number of sheets.

One More Example

• The cube graph Q3 is a two fold cover over the complete graph K4.

• The vertex fibers are composed of pairs of antipodal vertices.

Covers over Pregraphs

• K4 can be understood as a four-fold cover over a pregraph on one vertex (one loop and one half-edge).

Voltage Graphs

• X = (V,S,i,r) – connected (pre)graph.

• (,A) – permutation group acting on space A.

• :S – voltage assignment.

• Condition: for each s 2 S we have [s][r(s)] = id.

Voltage Graph Determines a Covering Graph

• Each voltage graph (X,,A,) determines a covering graph Y with covering projection f: Y X as follows:

• Covering graph Y = (V(Y),S(Y),i,r)• V(Y) := V(X) x A• S(Y) := S(X) x A• i: S(Y) V(Y): i(s,a) := (i(s),a).• r: S(Y) S(Y): r(s,a) := (r(s), [s](a)).

• Covering projection f• f: V(Y) V(X): f(x,a) := x.• f: S(Y) S(X): f(s,a) := s.

• Sometimes we denote the covering graph Y by Cov(X;).

(Rhetorical) Questions

• “Different” voltage graphs may give rise to the “same” cover. What does the “same” mean in this context and how do we obtain all “different” voltage graphs?

• A voltage graph is determined in essence by the abstract group. What is the role of the permutation group?

• How do we ensure that if X is connected that Y is connected, too?

Kronecker Cover

• Let X be a graph. The canonical double cover or Kronecker cover, KC(X), is a twofold cover that is defined by a voltage graph that has nontrivial voltage from Z2

on each of its edges. It can also be described as the tensor product KC(X) = X £ K2.

Regular Covers

• Let Y be a cover over X. We are interested in fiber preserving elements of Aut Y (covering transformations).

• Let Aut(Y,X) · Aut Y be the group of covering transformations.

• The cover Y is regular, if Aut(Y,X) acts transitively on each fiber.

• Regular covers are denoted by voltage graphs, where the permutation group (, A) acts regularly on itself by left or right translations: (, ).

Dipole n

• A dipole n has two vertices joined by n parallel edges. We may call one vertex black, the other white. On the left we see 5.

• Each dipole is bipartite, that is why each cover over n is bipartite, too. 3 is cubic and sometimes called the theta graph .

Cyclic cover over a dipole – Haar graph H(n).

• H(37) is determined by the number 37, actually by its binary representation (1 0 0 1 0 1).

• k = 6 is the length of the sequence, hence we choose the group Z6.

• (0 1 2 3 4 5) – positions of “1”.• Positions of “1”s: 0, 3 and 5.

{0,3,5} are the voltages on . The corresponding covering graph is H(37).

0 3 5

Z6

Cages as Covering Graphs

• A g-cage is a cubic graph of girth g that has the least number of vertices.

• Small cages can be readily described as covering graphs.

1-Cage

• Usually we consider only simple graphs as cages. For our purposes it makes sense to define also a 1-cage. We define it to be the pregraph on the left.

• A 1-cage is the unique smallest cubic pregraph.

2-Cage

• The only 2-cage is the graph.

• We may view the 2-cage as the Kronecker cover over 1-cage.

11

Z2

K4, the 3-cage

• K4 is a Z4 covering over the 1-cage.

• In general, we obtain a Z2n covering over the 1-cage by assigning voltage 1 to the loop and voltage n to the half-edge.

• Exercise: What is the covering graph for the cover described above?

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K3,3, the 4-cage

• K3,3 is a Z6 covering over the 1-cage.

• It can also be seen as a Z3 covering over the 2-cage .

• Exercise: Express K3,3 as a covering graph over . Dtermine a natural number n, such that K3,3 is a Haar graph H(n).

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The Handcuff Graph G(1,1)

• By changing the voltage on the loop of the 1-cage we obtain a double cover G(1,1), the smallest generalized Petersen graph, known as the Handcuff graph.1

0

Z2

I graphs I(n,i,j) and Generalized Petersen graphs G(n,k)

• Cyclic covers over the handcuff graph are called I-graphs. Each I-graph can be described by three parameters n,i, and j, with i · j. In case i = 1 we call I(n,i,k) = G(n,k), the generalized Petersen graph.

• In particular, I(5,1,2) is the 5-cage.

i0

Zn

j

The 6-cage

• The 6-cage is the Heawood graph on 14 vertices. It is a 7-fold cyclic cover over the graph. But it is also a dihedral cover over the 1-cage.

• Let the presentaion of Dn be given as follows: Dn = <a,b|an,b2, ab=ba-1>

• Then the Heawood graph is the covering described on the left.

ba

D7

(3,1)-trees

• A (3,1)-tree is a tree whose vertices have valence 3 or 1 only.

• On the left we see the smallest (3,1)-trees I,Y and H.

(3,1)-cubic graphs

• A (3,1)-cubic graph is obtained from a (3,1)-tree by adding a loop at each vertex of valence 1.

• On the left we see the smallest (3,1)-cubic graphs I(1,1,1),Y(1,1,1,1) and H(1,1,1,1,1).

Coverings over (3,1)-cubic graphs

• By putting 0 on the tree edges and appropriate voltages on the loops of (3,1)-cubic graphs we obtain their Zn coverings.

• For the graphs on the left we obtain the I-graphs, Y-graphs and H-graphs: I(n,i,j),Y(n,i,j,k) and H(n,i,j,k,l).

Zni

i

i

j

j

j

k

k

l

Covers Determined by Graphs

• We know already that there exists a cover, namely the Kronecker cover, that depends only on X itself and the voltage assignment plays a minor role.

• Now we will present some covers possessing a similar property.

Coverings and Trees

• Let X be a connected graph and let Cov(X) denote all connected covers over X:

• Cov(X) = {(Y,)| Y connected and : Y ! X, covering projection}. For each connected X we have (X,id) 2 Cov(X).

• Proposition: For a connected X we have Cov(X) = {(X,id)} if and only if X is a tree.

• This fact holds both for finite and locally finite trees.

Universal cover

• Let X, Y and Z be connected graphs and let : Y ! X and :Z ! Y be covering projections.

• Let us consider the class Cov(X) of all coverings over X. We may introduce a partial order in Cov(X). (Y,) < (Z,) if there exists a covering projection (Z,) 2 Cov(Y) so that = .

• Proposition: Any connected finite or locally finite graph X can be covered by some tree T; : T ! X.

• Proposition: Any connected finite or locally finite graph X can be covered by at most one tree T.

• Proposition: Let : T ! X be a covering projection from a tree to a connected graph X. Then for each covering : Y ! X there exists a covering : T ! Y such that = .

• Corollary: For each connected X the poset Cov(X) has a maximal element (T,) where T is a tree.

• The maximal element (T,) 2 Cov(X) is called the universal covering of X.

Construction of Universal Cover

• There is a simple construction of the universal covering projection.• Let X be a connected graph and let T µ X be a spanning tree.

Furthermore, let S = E(X) \ E(T) be the set of edges not in tree T. • Consider S to be the set of generators for a free group F(S) and let

F(S) be the voltage group.• Let us assing voltages on E(X) as follows:

• If e 2 E(T) the voltage on e is the identity.• If e 2 S the voltage is the corresponding generator (or its inverse)

• Note: The construction does not depend on the choice of edge directions.

• Proposition: The described construction yields the universal cover.

Examples

• Example: The universal cover over any regular k-valent graph is a regular infinte tree T(1,k).

Valence Partition and Valence Refinement

• Let G be a graph and let B = {B1, ..., Bk} be a partition of its vertex set V(G) for which there are constants rij, 1 · i,j · k such that for each v 2 Bi there are rij edges linking v to the vertices in Bj. Let R = [rij] be the corresponding k £ k matrix. B is called valence partition and R is called valence refinement. If k is minimal, then B is called minimal valence partition and R is called minimal valence refinement.

• Two refinements R and R’ are considered the same if one can be transformed into the other by simultaneous permutation of rows and columns.

• A refinement is uniform, if each row is constant.

Construction

• Given graphs G and G’ with a common refinement.• Let mij denote the number of arcs in G of type i ! j.• Let ni denote the number of vertices in G of type i.• Let bij = lcm(mij)/mij. (If mij = 0 , let bij undefined).• Let ai = lcm(mij)/ni.• Note that bij and ai depend only on the common matrix R and are the same for

both graphs G and G’.• Let l(e) or l(e’) be a linear order given to all type i ! j arcs with a common

initial vertex i(e) (or i(e’)).• Let V(H) = {(i,v,v’,p)|v and v’ of type i, p 2 Zai}• Let S(H) = {(i,j,e,e’,q)|e and e’ of type i ! j, q 2 Zbij}• r(i,j,e,e’,q) := (j,i,r(e),r(e’),q)• i(i,j,e,e’,q) := (i,i(e),i(e’),q rij + l(e)-l(e’)} • H is a common cover of G and G’.

Computing Minimal Valence Refinement

• Let r[u,B] denote the number of edges linking u to the vertices in B.• Algorithm [F.T.Leighton, Finite Common Coverings of Graphs,

JCT(B) 33 1982, 231-238.] • Step 1. Place two vertices in the same block if and only if they have

the same valence.• Step 2. While there exist two blocks B and B’ and two distinct vertices

u,v in B with r[u,B’] r[v,B’] repeat the following:• Partition the block B into subblocks in such a way that two vertices u,b of B

remain in the same block if and only if r[u,B’] = r[v,B’] for each B’ of the previous partition.

• Step 3. From the minimal valence partition B compute the minimal vertex refinement R.

• Note: We may maintain R during the run of the algorithm as a matrix whose elements are sets of numbers.

Comon Cover• Theorem. Given any two finite graphs G and H, the

following statements are equivalent: 1. G and H have the same universal cover,2. G and H have a common finite cover,3. G and H have a common cover,4. G and H have the same minimal valence refinement.5. G and H have the same some valence refinement. • Homework. Find the result in the literature and

construct a finite common cover of G(5,2) and G(6,2).

Petersen graph

• An unusual drawing of the Petersen graph.

Petersen graph G(5,2) and graph X.

Kronecker Cover - Revisited

• The Kronecker cover KC(G) is an example of a cover determined by the graph itself.

• Exercise. Show that G(5,2) and X have the same Kronecker cover.

THE covering graph

• Let G be a graph with the vertex set V. By THE(G) we denote the following covering graph.

• To each edge e = uv we assign the transposition e = (u,v) 2 Sym(V). The resulting covering graph has two components, one being isomorphic to G. The other component is called THE covering graph.

Examples

• On the left we see THE(K2,2,2).

• The construction resembles truncation.

• Each vertex is truncated and an inverse figure is placed in the space provided for it.

• Theorem: If G is planar, then THE(G) is planar.

The fundamental group of a graph.

• Let G be a connected graph rooted at r 2 V(G) and let denote the collection of closed walks rooted at r.

• Let and be two closed walks rooted at r. The composition is also a closed walk rooted at r.

• We may also define -1 as the inverse walk.

• Finally, we need equality (equivalence). • 1 2 ~ 1 e e-1 2.

• (G,r) := /~ is a group, called the fundamental group of G (first homotopy group).

• Fact: (G,r) is a free group on m-n+1 generators.

The first Homology group of a graph

• Let G be a connected graph and T one of its spanning trees. Each edge h 2 G\T not in T defines a unique cycle C(h) µ E(T) [ h.

• The charactersitic vector h 2 {0,1}m, h(e) = 1, if e 2 C(h) and h(e) = 0, represents C(h). The set of all charactersitic vectors spans a m-n+1 dimensional Z-module in Zm. This can be also viewed as a free abelian group isomorphic to Zm-n+1.

• This group is called the first homology group H1(G,Z). We may replace Z by Zk and obtain the first Zk homology group Zk

m-n+1.

Pseudohomological Covers

• Idea: Let G be a graph, T one of its spanning trees and H = {h1,h2,...,hm-n+1} = E(G)\E(T). Let (H) be a group with m-n+1 interchangeable generators H. The pseudohomological -cover HOM(G,) is determined by a voltage graph with (e) = id, for e 2 E(T) and (h) = h, for h 2 E(G)\E(T).

• Main Question. Is HOM(G,) independent of the choice of T and the selection of the generators or their inverses? If the answer is yes, the covering is called homological cover.

Pseudohomological 2-cover

• Let G be a graph and T a spanning tree of G. The pseudohomological 2-cover HOM(G,Z2,T) is determined by a voltage graph with (e) = 0, for e 2 E(T) and (e) = 1, for e E(T).

• Theorem. If G is connected then HOM(G,Z2,T) is connected if and only if G is not a tree.

Example

• The two voltage graphs on the left determine different pseudohomological Z2 covers.

• Cov(G,2) is bipartite and Cov(G,1) is not.

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Switching

• Let (G,) be a voltage graph. Let : V(G) ! be an arbitrary mapping, called switching, that assigns voltages to vertices. Define a new voltage assignment as follows:

• (s) := (i(s)) (s) (i(r(s))-1.• is well-defined.• Namely (r(s)) = (i(r(s))) (r(s)) (i(s))-1.• Hence (r(s))-1 = (i(s)) (r(s))-1 (i(r(s)))-1 = (i(s)) (s) (i(r(s)))-1 =

(s).• Clearly for any switching the graphs Cov(G,) and Cov(G,)

coincide.• Given (G,) and any spanning tree T. There exists a switching such

that the resulting is the identity on T. • If, in addition, T is rooted at v, we may select (v) = id (or arbitrarily)

and this determines switching completely.

Homological Elementary Abelian Covers

• Let G be a graph with a spanning tree T. Let k = m-n+1 be the number of edges in G\T. Define the voltage assignment such that each non-tree edge gets the voltage ei = (0,0,..,0,1,0,...,0) 2 Zp

k.• Claim: If p is prime, then Cov(G,) is

independent of T.• Question: What happens in the case p is not

prime?

Tree-To-Tree Switch

• Let T and T’ be two spanning trees of G. Let H = {h1, h2, ..., hk} be the co-tree edges of T. Let r be the root of G. For each vertex w 2 V(G) there is a unique path P(T’,w,r) on the three T’ from w to v. Let S(w) µ H be the collection of co-tree edges on this path. Let S(w) be the label given to w. Hence (w) = { hi| hi 2 S(w)}.

• Claim: Starting with a homological voltage assignment relative to T and applying the tree-to-tree switch , the voltages are given as follows:

• The edges on T’ get voltage 0.• An edge e = uv on a co-tree T’ get the voltage:

• k(e) = S(u) + S(v) if e 2 T.• k(e) = S(u) + S(v) + h(e) if e T.

• Each co-tree edge e defines a cycle C(e). The net voltage on C(e) is equal to k(e).

• The voltages k(e), for e T’ span the whole Z2k.

Real Homological Cover

• Let G be a graph with a given cycle basis C1, C2, ..., Ck. Direct each cycle and assign to each edge of Ci the voltage ei 2 Zn

k. The final voltage assignmnet is given by adding the partial voltages.

• An example is given on the left. The cycle basis is determined by a spanning tree.

(0,1)

(1,0)

(0,1)

(1,1) (1,0)

Z22

Least Common Cover

• Theorem: There exist finite connected graphs H1, H2, G1, G2 such that G1 and G2 are both double covers of H1 and H2.

• Proof. We start with graphs G = G(5,2) and X that we know from earlier.

G+X and G + G

• Given two graphs G and H we form G+H by adding an edge between them.

• On the left we see G + X and G + G.

• The resulting graph depends on the choice of endpoints of the added edge.

H1 and H2

• Define H1 and H2 as follows:

• H1 = G + X + X and H2 = G + G + X.

Covers of G+H.

• A double cover of G+H can be split into two double covers G* and H* which are then joint by a pair of edges. We denote the resulting graph by G* ++ H*.

• For instance KC(G + X) = KC(G) ++ KC(X) = G(10,3) ++ G(10,3).

End of Proof

• Let G1 = G(10,3) ++ G(10,3) ++ G(10,3) and G2 = G(10,3) ++ G(10,3) ++ 2X.

• G1 and G2 are distinct. They are both covers of H1 and H2.

Exercises 4-1

• N1: Prove that each double sheeted cover is regular.

• N2: Find an example of a three sheeted cover that is not regular.

• N3: Express the graph on the left as a 6-fold cover over a pregraph on a single vertex.

Exercises 4-2

• Let Znk be an elementary abelian group. Let

S be a set of generators with the following property. Each element is a 0-1 vector. They generate the whole group.

• N4. Show that |S| = k.

• N5. Show that there is an automorphism of the group mapping S to the standard generating set.

Exercises 4-3

• The graph on the left is called the Heawood graph H. Prove:– N6. H is bipartite.– N7. H is a Haar graph

(Determine n, such that H = H(n))

– N8. Express H as a cyclic cover over .

– N9. Show that there are no cycles of lenght < 6 in H.

– N10. Show that H is the smallest cubic graph with no cycles of length < 6.

Exercises 4-4

• N11. Express the 7-cage as a covering graph.

• N12. Express the 8-cage as a covering graph.

Homework 4-1

• H1: Prove that the Kronecker cover is bipartite.• H2: Prove that generalized Petersen graph G(10,2)

is a twofold cover over the Petersen graph G(5,2).• H3: Determine the Kronecker cover over G(5,2).

• H4: Determine a Zn covering over the handcuff graph G(1,1), that is not a generalized Petersen graph G(n,r).

Homework 4-2

• H5. Given a connected graph G with n vertices and e edges and with valence sequence (d1, d2, ..., dn). Determine the parameters for THE(G).

• H6. Determine all connected graphs G for which girth(G) girth(THE(G)).

5. Symmetry in Graphs

The symmetries of a graph are the elements of its automorphism group. In particular, the symmetry of a graph does not depend on its drawing. Sometimes it is possible to draw a graph to show all itsgraph symmetries, and sometimes not. For example, the most symmetric drawing of K_5 has only 10 symmetries, however the graph K_5 has 5! = 120 automorphisms.

Aut G revisited.

• Recall that the automorphism group Aut G for a simple graph G can be viewed as a subgroup of Sym(V(G)) or a subgroup of Sym(E(G)).

Example for Aut G acting on V(G).

• |Aut G| = 4 .• V(G) ={1,2,3,4} • Id = (1)(2)(3)(4)• = (1)(3)(2 4)• = (1 3)(2)(4)• = = (1 3)(2 4)

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Example for Aut G acting on E(G).

• |Aut G| = 4.• EG ={a,b,c,d,e} • Id = (a)(b)(c)(d)(e)• = (a d)(b c)(e)• = (a b)(c d)(e)• = = (a c)(b d)(e)

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Induced Action on E(G)

• For a simple graph G the action of Aut G on V(G) induces an action of Aut G on E(G).

• For example: since a = 1 ~ 2 and (1) = 1, (2) = 4, we have (a) = 1 ~ 4 = d.

Example for Orbits

• |Aut G| = 4• V(G) ={1,2,3,4} is

partitioned into two orbits R = {1,4} and S={2,3}.

• E(G) = {a,b,c,d,e} has two orbits: Z = {a,b,e,d} and M={c}.

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Cayley Table for the dihedral group Dih(3) = D3.

1 X X2 Y XY X2Y

1 1 X X2 Y XY X2Y

X X X2 1 XY X2Y Y

X2 X2 1 X X2Y Y XY

Y Y X2Y XY 1 X2 X

XY XY Y X2Y X 1 X2

X2Y X2Y XY Y X2 X 1

Cayley Color Digraph

• Information in Cayley table is redundant!

• Two possibilities:– Left Cayley graph (will

not be used )

– Right Cayley graph.

v

(v)

(v)

LEFT

v

(v)

(v)

RIGHT

Cayley Color Digraph for D3.

• Right Cayley Color Digraph

• Convention: Since 1 = Y2 we may use the undirected version of the edge..

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X2Y

X Y

Cayley Graph (Right)

• Let be a group and ½ a set of generators, such that:

• Symmetric: = -1

• Does not contain identity: 1 .

• To a pair (,) we can associate a Cayley graph X = Cay(,) as follows:

• V(X) = • g ~ h , g-1h 2 .

Basic Theorem about Cayley graphs

• Graph X is a Cayley graph, if and only if there exists a subgroup · Aut X, acting regularly on V(X)!

• Exercise: Prove that the Petersen graph is not a Cayley graph.

Direct Product

• The Cayley graph of a direct product corresponds to the Cartesian product of Cayley graphs.

• Problem: Define the free product of groups and explore the corresponding product construction of rooted Cayley graphs.

Frucht’s Theorem

• Theorem: For each finite group there exists a graph X, such that isomorphic to Aut X.

Vertex-Transitive Graphs

• If group acts on a space V with a single orbit ([x] = V), we say that the action is transitive.

• Let (,V) denote a permutation group acting on V. Let [x] be any of its orbits. The restriction (,[x]) is transitive.

Vertex Transitvity

• Graph X is vertex transitive, if Aut X acts transitively on V(X).

• Example: Three out of the four graphs on the left are vertex transitive.

• Question: Which Generalized Petersen graphs G(n,r) are vertex transitive?

Vertex Transitvity and Regularity

• Proposition: Each vertex transitive graph is regular.

• Proof: If an automorphism maps vertex u to vertex v, then deg(u) = deg(v). Hence all vertices of an orbit have the same valence. A vertex transtive graph has a single vertex orbit, therefore deg(v) is constant and the graph is regular.

Vertex-Transitive Subgraphs

• Let G be a graph and [x] ½ V(G) an orbit of Aut G. The induced subgraph <[x]> is vertex transitive.

• Let H ½ G be an induced subgraph of G. Let < Aut H be the group of those automorphisms that can be extended to the group of automorphisms of G.

• Given H and given < Aut H. Find a graph G, such that H is induced (isometric, convex) in G.

Edge Transitive Graphs

• Graph X is edge transitive, if Aut X acts transitively on E(X).

• On the left we see antiprisms A7, A3, the Möbius ladder M4 and the prism 6. Which of these graphs are edge transitive?

Edge Transitive Graphs that are not Vertex Transitive

• Theorem: An graph X which is edge transitive but not vertex transitive must be bipartite.

• Lemma: If both endvertices of an edge transitive graph belong to the same orbit, the graph is vertex transitive.

• Lemma: An edge transitive graph has at most two vertex orbits.

• Lemma: If an edge transitive graph has two vertex orbits, each of them is an independent set.

Arc Transitive Graphs

• Graph X is arc transitive, if Aut X acts transitively on the set of arcs S(X).

• Example: G(5,2) is arc transitive, P3 is not.

Arc and Edge Transitivity

• Proposition: Any arc transitive graph X is edge transitive.

• Proof: Take any edges e and f. Each of them has two arcs e+ , e- and f+ , f-. Since X is arc transitive, there exists an automorphism 2 Aut X, mapping e+ to f+: (e+ ) = f+. Therefore it maps e- to f-: (e- ) = f-. We conclude that (e) = f.

Arc and Vertex Transitivity

• Theorem: An arc transitive graph X without isolated vertices is vertex transitive.

• Proof. Take any vertices u and v. Since they are not isolated there are arcs e and f such that i(e) = u and i(f) = v. Since X is arc transitive there exists an automorphism 2 Aut X, mapping e to f. By definition it maps u to v.

Arc Transitive I-graphs

• The only arc transitive I-graphs are the seven generalized Petersen graphs: G(4,1), G(5,2), G(8,3), G(10,2), G(10,3), G(12,5), G(24,5).

Arc-transitive Y graphs

• Horton and Bouwer showed in 1991 that the only arc-transitive Y graphs are Y(7,1,2,4), Y(14,1,3,5) (girth 8), Y(28,1,3,9) (girth 8) and Y(56,1,9,25) (girth 12).

Vertex and Edge Transitivity.

• Proposition: There exists a graph X, that is vertex transitive, but not edge transitive.

• Proposition: There exists a graph X, that is edge transitive, but not vertex transitive.

Arc-transitive H graphs

• There are only two arc-transitive H graphs: H(17,1,2,4,8) and H(34,1,9,13,15) (girth 12).

Arc-transitive (3,1)-cubic graphs

• There is a complete characterization of arc-transitive connected (3,1)-cubic graphs.

• 7 – I-graphs• 4 – Y-graphs• 2 – H-graphs• Exercise: Prove that if the connectivity

condition is dropped the number of arc-transitive graphs is infinite.

s-Arc-Transitive Graphs

• An s-arc in a graph X is a sequence (a0,a1, ..., as) of vertices of X such that aiai+1 is an edge in E(X) and ai-1 ai+1.

• A graph X is s-arc-transitive if its automorphism group acts transitively on the set of its s-arcs and does not act transitively on the set of its (s+1)-arcs.

1/2-Arc-Transitive Graphs

• A vertex-transitive graph X that is edge-transitive but not arc –transitive is called ½-arc-transitive graph.

Vertex, Edge and Not-Arc Transitvity

• Theorem: There exist vertex- and edge- transitive graphs that are not arc-transitive.

• The Holt graph on the left is the smallest such example. It has 27 vertices and is 4-valent.

Holt graph - Revisited

• 4-valent Holt graph H is a Z9-covering over the graph on the left.

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-2

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Z9

Half Arc Transitive Graph

• There are several families of ½-arc-transitive graphs (many discovered by mathematicians in Slovenia).

• Theorem: Each ½-arc-transitive graph is 2k-regular for some natural number k.

• Proof: Half arc transitive action on X means an action on S(X) with two equaly sized orbits. For each s 2 S(X) the orbits [s] and [r(s)] are different. No edge may be mapped to itself by an automorphism without fixing both of its endvertices. This implies that giving direction to one edge implies directions in every other edge. Aut X acts transitively on such directed edges.

• If we have at any vertex v the inequality indeg(v) > outdeg(v), the same inequality would hold at every vertex. This contradicts the well-known fact:

• indeg(x) = outdeg(x).

LCF Notation for Cubic Graphs

• A cubic graph X on 2n vertices, with a given Hamilton cycle, can be easily encoded by successive lengths of the cords along the Hamilton cycle.

• Example: For the graph on the left:

• LCF[3,4,2,3,4,2] = LCF[3,-2,2,-3,-2,2]

LCF – Example

• Let us introduce simple notation (by example):

• (a,b,c)2 = (a,b,c,a,b,c)

• (a,b)-2 = (a,b,-b,-a)2

• Example: LFC[(3,-3)4] = LCF[(3)-4] = Q3.

Heawood Graph - LCF

• LCF[(5)-7] denotes the Heawood graph.

Edge Orbits of a Vertex-Transitive Graph.

• Theorem: In a vertex-transitive graph X of valence d the number of edge orbits · d.

• Proof: Let i(e) = v, hence the arc e has endpoint v. Each vertex u has at least one arc f, with i(f) = u and vertex-transitivity implies [f] = [e]. Around vertex v there are at most d edge orbits passing by automorphism from vertex to vertex. This way we exhaust all edges and therefore their orbits.

Regular action of Aut X.

• Definition: A vertex-transitive graph X, such that |Aut X| = |V(X)| is called a graphical regular representation (GRR) of its automorphism group = Aut X.

• Remark: If Aut X acts transitively on V(X), it does not mean that there exists a subgroup · Aut X, acting on V(X) regularly.

0-Symmetric Graphs

• Definition: A vertex-transitive cubic graph X with three edge orbits is 0-symmetric.

• Theorem: The class of cubic graphs, that are GRR coincides with the class of 0-symmetric graphs.

• Proof: Use Lemma on orbits and stabilizers and two other lemmas.

Two Lemmas

• Let X be a graph and a group of automorphisms. The stabilizer x of vertex x acts on the set of neighbors of x: X(x).

• Lemma: In a vertex transitive graph the number of edge orbits equals the number of orbits of the action of x on X(x).

• Lemma: The only permutation group acting faithfully and fixing all elements of a space is trivial.

Examples

• Each 0-symmetric graph is a Haar graph.

• The smallest example is H(9;S) = H(28 + 27 + 25), where S = {0, 1, 3}.

• LCF[{5,-5}9].

The Mark Watkins Graph

• The smallest 0-symmetric Haar graph H(n;{0,a,b}) with the property that gcd(a,n) > 1, gcd(b,n) > 1,gcd(b-a,n) > 1, and gcd(a,b) = 1, has parameters n = 30, a = 2, b = 5. It is called the Mark Watkins graph.

Semi-Symmetric Graphs.

• Definition: A regular graph X which is edge transitive, but not vertex transitive, is called semi-symmetric.

• On the left we see one of them, the 4-valent Folkman graph.

Direct Product of Groups - Revisited.

• A £ B – direct product of groups A and B, defined on the cartesian product of the elements. Group operation by components.

• Example. Z3 £ Z3 has 9 elements: (0,2) + (1,2) = (1,1).

• Finite abelian groups $ (finite) direct products of (finite) cyclic groups.

6. Symmetry in Metric Spaces

• Let (M,d) be a metric space.• Iso(M) is the group of isometries.• Sim1(M) is the group of similarities of type 1.• Sim2(M) is the group of similarities of type 2.• Let B(a,r) = {x 2 M|d(a,x) · r} be the ball centered at a

with radius r.• Let S(a,r) = {x 2 M|s(a,x) = r} the sphere centered at a

with radius r.

Isotropic Metric Spaces

• A metric space (M,d) is said to be isotropic at point x 2 M, if all spheres S(x,r) centered at x are homogeneous. M is said to be isotropic, if it is isotropic at each of its points.

Homogeneous Metric Spaces

• A metric space (M,d) is said to be homogeneous, if all points are indistinguishable, i. e. if Iso(M) acts transitively on the points.

• For connected graphs the above condition is equivalent to being vertex-transitive.

Some Results

• Claim 1. Every sphere of an isotropic space is homogeneous.

• Let X ½ M. • Iso(M,X) is the group of isometries fixing X set-wise.• Iso(M;rel X) is the group of isometries fixing X point-

wise.• Iso(X) are the isometries of X.• S(X) is the set of isometries of X that can be extended to

isometries of M.

Distance Set

• Let (M,d) be a metric space and let x 2 M. Let D(x) = {d 2 R+| d(x,v), v 2 M}. D(x) is called a distance set at x. M is said to have constant distance set if D(u) = D(v) for any pair of points u,v 2 M.

Distance-Transitive Metric Spaces

• A metric space (M,d) is said to be distance-transitive if for any four points a,b,p,q 2 M with d(a,b) = d(p,q) there exists an isometry h of M, mapping a to p and b to q.

• Theorem. (M,d) is distance-transitive if and only if it is homogeneous and isotropic.

• Note: There are isotropic non-homogeneous metric spaces.

Distance-Transitive Graphs

• Connected graph G is also a metric space. We may speak of isotropic graphs and distance-transitive graphs.

• For instance Km,n is isotropic but not distance-transitive.

Cubic Distance-Transitive Graphs

• Theorem: There are only 12 cubic distance-transitive graphs:

1. 4, nonbipartite, girth = 3, K4

2. 6, bipartite, girth = 4, K3,3

3. 10, nonbipartite, girth = 5, G(5,2)

4. 8, bipartite, girth = 4, Q3

5. 14, bipartite, girth = 6, Heawood

6. 18, bipartite, girth = 6, Pappus

7. 28, nonbipartite, girth = 7, Coxeter

8. 30, bipartite, grith = 8, Tutte 8-cage

Cubic Distance-Transitive Graphs

• Theorem: There are only 12 cubic distance transitive graphs:

09. 20, nonbipartite, girth = 5, G(10,2)

10. 30, bipartite, girth = 6, G(10,3)

11. 102, nonbipartite, girth = 9, Biggs –Smith H(17:1,2,4,8)

12. 90, bipartite, grith = 10,Foster

Example: Foster Graph

• The bipartite Foster graph on 90 vertices is the largest cubic distance-transitive graph.

• LCF[{17,-9,37},-15]

Biggs-Smith Graph

• The Biggs-Smith graph H(17;1,2,4,8) has 102 vertices and girth 9.

Biggs-Smith Graph

• The Biggs-Smith graph H(17;1,2,4,8) has 102 vertices and girth 9.

• Its Kronecker cover is bipartite and has girth 12.

Odd graph On.

• Vertex set: all n-1 subsets of a 2n-1 set:

• |V(On)| = C(2n-1,n-1).

• Two sets are adjacent if they are disjoint.

• Valence: n.

• O2 = K3

• O3 = G(5,2)

• O4 = Gewirtz graph.

Quartic Distance-Transitive Graphs

• Theorem: There are only 15 quartic distance transitive graphs:

1. K5

2. K4,4

3. L(K4)

4. L(K3,3)5. L(G(5,2))

Quartic Distance Transitive Graphs

1. L(Heawood)

2. K2 £ K5

3. Heawood[3].4. (4,6) cage

5. Gewirtz graph O4.

Quartic Distance-Transitive Graphs

1. L(Tutte8cage)

2. Q4

3. 4-fold cover of K4,4

4. (4,12) cage

5. K2 £ O4.

Hamiltonicity

• Most vertex-transitive graphs have Hamilton cycles.

• There are only 4 known vertex-transitive graphs without Hamilton cycle. [All four of them have a Hamilton path.]

Exercises 6-1

• N1. Find an isotropic metric space that is not homogeneous.

• N2: Prove that G(n,k) is vertex transitive, if and only if k2 § 1 mod n, or else n=10 and k=2.

• N3: Prove that Cn, Kn, Qn are all vertex transitive.

• N4: Which complete multipartite graphs Ka,b, Ka,b,c, ... are vertex transitive?

• N5: Prove that the Cartesian product of vertex transitive graphs is vertex transitive.

Exercises 6-2

• N6: Write an LCF code for the Dürer graph.

• N7: Write an LCF code for K4.

• N8: Write an LCF code for M3 = K3,3. Generalize to the Möbius ladder Mn.

Exercises 6-3

• N9: Prove that Z3 £ Z3 À Z9.

• N10: Prove that Z2 £ Z3 Z6.

• N11(*): Prove that any finite abelian group A is isomorphic to the direct product A(n1,n2,...,nk) = Zn1

£ Zn ... £ Znk, where n1|n2|...|nk.

• N12(*): Prove that the groups A(n1,n2,...,nk) and A(m1,m2,...,mj) with n1|n2|...|nk and m1|m2|...|mj are equal if and only if j= k and nt=mt, for each t.

Homework 6-1

• H1. Find a better drawing of Gewirtz graph.

Homework 6-2

• H2. Find the definition and a drawing of any missing quartic graph in the previous theorem.

• H3. Determine all groups that have a cycle Cn their Cayley graph.

7. Representations of Graphs and their Symmetries

• Graph drawings in a vector space or on a surface may be considered graph representations.

• Graphs obtained from applications may have interpretations for their vertices and edges, yielding various graph representations.

Representation of Graphs• Let G be a graph and let V be a set. A pair of mappings• V:V(G) ! V and• E:V(G) ! P(V) is called a • V-representation of graph G • if for any edge e = uv 2 E(G) • we have {V(u),V(v)} µ E(uv). • If there is no danger of confusion we will drop the

subscripts and denote both mappings simply by .

Representation of Graphs• Usually we require V to be a vector space (this is what

C. Godsil and G. Royle do in their book Algebraic Graph Theory, Springer, 2001). But that is not always the case.

• In their definition Godsil and Royle use a single mapping defined on the vertices.

• In such a case we may extend the mapping on the edge set in an arbitrary way, for instance by taking E(uv) := {V(u),V(v)}. [This works for all representations.]

Graph Representation Examples

• For the cube graph Q3 there are several useful representations:

• [3 dimensional real representation] In R3 the eight vertices are mapped to the eight points of {0,1}3.

• The two drawings of Q3 in the Euclidean plane can be interpreted as representations in

• [2 dimensional real representation] R2 or in • [1 dimensional complex representation] C.

• In the latter case, the points in the complex plane are given by {eik|0 · k · 7}.

Edge Representation

• Let e = uv 2 E(G). For a representation in Rn or Cn it makes sense to define

• (e) = conv(V(u),V(v)). • Hence an edge is represented as the

segment joining the two endpoints.• For other representation spaces a different

edge representation may be of interest.

(u)

(v)

Edge Extensions• Let e = uv 2 E(G).• There are several possible edge

extensions: • (e) = {(u),(v)}. • (e) = {(u),r,(v)}.

• r = ((u)+(v))/2.

• (e) = conv((u),(v)). • (e) = aff((u),(v))• We may speak of barycentric,

convex and affine edge extensions, respectively.

• But there are several other interpretations of r and a variety of possible edge extensions.

(u)

(v)

(u)

(v)

r

(u)

(v) (v)

(u) (u)

(v)

r

Graph Representation vs. Graph Drawing

• There is some overlap but there are many differences.• In graph drawing (in the broad sense of the word) the object is to

find algorithms to draw a graph (usually in the plane) with certain restrictions or with some optimization criterion. [Computer Science Approach.] See for example: Annotated bibliography on graph drawing algorithms, by Di Battista, Eades, Tamassia and Tollis.

• In graph representation we label vertices (= add coordinates). We may look at this as a functor from the category of graphs to the category of coordinatized graphs. [Mathematical Approach].

• We will use the word graph drawing in a narrow sense of the word (as a special representation).

One Dimensional Real Representation

• Let : V(G) ! R be a graph representation. Then such a representation is sometimes called a fitness landscape over G and is a cost function or potential function.

• For many problems of combinatorial optimization we may model a solution space by a graph. To each vertex we assign a cost and try to find a feasible solution yielding a minimum cost.

Nodal Domains

• A one dimensional representation defines a partition of the vertex set into three classes: V+, V-, and V0.

• V+ = {v 2 V(G)| (v) > 0}.• V- = {v 2 V(G)| (v) < 0}.• V0 = {v 2 V(G)| (v) = 0}.• A nodal domain is a

connected component of the graph induced by V+or V-. [Weak nodal domain V+ [ V0].

Local Search

• Idea: Feasible solutions form a metrc space. For any solution x choose a neighborhood N(x).

• In N(x) find a solution x’ with smaller cost value: (x’) < (x).

• Repeat the procedure as long as possible.

r-Neighborhoods

• Y – feasible solutions form a metric space.

• Select radius r > 0.• Declare: • N(x) := B(x,r) = {y 2 Y | d(x,y) ·

r}

• (Y,d,r) defines a graph G:• V(G) := Y

• x ~ y if and only if d(x,y) · r.

x

N(x)

Neighborhoods in Graphs

• For graphs we usually take N(x) the set of neighboors of x.

• For a connected graph this is equivalent to

• N(x) = B(x,1) = {y 2 Y | d(x,y) · 1}

x

N(x)

Local Search Algorithm

Input: Connected graph G, representation : V(G) ! R, initial value x0 2 V(G).

• Output: Local minimum at x.• x à x0;• Repeat:

• finish à true;• For each y 2 N(x) if (y) < (x) then

» x à y; finsh à false

• Until finish

Simulated Annealing - Idea

• Use local search to move downhill ( < 0).

• Possible temperature dependent jumps uphill ( > 0).

• Lower the temperature.

(x)

(y)

Characteristic Vector

• Let G be a graph and S µ V(G) a set of vertices. Let GS denote the induced subgraph of G, defined by set S.

• The characteristic vector

• S: V(G) ! {0,1} with• S(x) = 1 if x 2 S and

• S(x) = 0 otherwise,

• is a graph representation.

Vertex Coloring as Graph Representation

• Let : V(G) ! {1,2,...,k} be a representation of graph G and for each edge let (uv) = {(u),(v)}.

• Then is a proper vertex coloring if and only if for each edge |(uv)| = 2.

Identity Representation

• Each graph G has a trivial, or identity representation:

• V = id: V(G) ! V(G).

• E: uv {u,v}.

• For simple graphs, E is an injective map.

Point Configuration

• A point configuration S µ V is a collection of elements of some space V. Usually we consider point configurations in R2.

• If is a V-representation of G then the image S = (V(G)) is a point configuration.

• We say that is vertex faithful is :V(G) ! S is a bijection. We are mostly interested in vertex faithful representations.

Graph Construction from Point Configurations

• Given point configuration S µ V and a symmetric predicate

• : V ! {true,flase}• (p,q) = (q,p).• We define a graph:• G = G(S,) as follows:• V(G) := S.• x ~ y if and only if (x,y) =

true.• Example:

• V = Rn,• (x,y) = (d(x,y) · r)

• Let be a “nice” topological space such as metric space and G be a general graph. A mapping :G is defined as follows:

1. Injective mapping :V(G)

2. Family of continuous mappings e:[0,1] for each edge e = uv so that e( 0) = (u) and e(1) = (v).

3. In the interior of the interval e is injective.

• Each embedding would qualify. For embeddings we need more!

• Note that defines a representation of G in .

Embeddings are Representations

• Think of K3 ¤ K3 embedded in a torus, the torus, in turn, is embedded in R3. We obtain a representation of this graph in the torus and another one in R3.

New Representations from Old

• We are investigating techniques that use existing representations for obtaining new, more sophisticated representations.

Stereographic Projection

• There is a homeomorphic mapping of a sphere without the north pole N to the Euclidean plane R2. It is called a stereographic projection.

• Take the unit sphere x2 + y2 + z2 = 1 and the plane z = 0.

• The mapping p: T0(x0,y0,z0) T1(x1,y1) is shown on the left.

N

T0

T1

Stereographic Projection

• The mapping p: T0(x0,y0,z0) T1(x1,y1) is shown on the left.

• r1 = r0/(1-z0)

• x1 = x0/(1-z0)

• y1 = y0/(1-z0)

N

T0

T1

Stereographic projection and representations

• We may use stereographic projection to get an R2 drawing from an R3 drawing.

• Note that the representation of edges is computed anew!

Example

• Take the dodecahedron and a random point N on a sphere.

• The resulting stereographic projection is depicted below.

• A better strategy is to take N to be a face center.

Example

• A better strategy is to take N to be a face center as shown on the left.

• Only vertices are projected. The edges are re-computed.

Representation of Graphs in Metric Space

• Sometimes we may take V to be a metric space, projective space or some other structure.

• If (V,d) is a metric space we may define the energy of the representation .

• Ep() = [uv 2 Ed((u),(v))p]1/p, for 1 · p · 1.

Euclidean metric in Rn.

• The set of real n-tuples• Rn := {x = (x1,x2,...,xn)|xi 2 R, 1 · i · n} • carries a number of important mathematical

structures. The mapping • dp(x, y) = [(x1 – y1)p + (x2 – y2)p + ... + (xn – yn)p]1/p.

• makes (Rn,dp) a metric space for 1 · p · 1.• For p = 2 the usual Euclidean metric is obtained.• There are important and deep results by László

Lovász et al.

Metric Space - Revisited

• If (M,d) is a metric space, then for any A µ M with the induced metric (A,d) is also a metric space, a subspace.

• A natural question is: when are two metric spaces (M,d) and (M’,d’) considered isomorphic?

• There are two types of mappings that are candiates for “isomorphism”.

Isometries

• Let (M,d) and (M’,d’) be two metric spaces. A bijective mapping : M ! M’ is called is isometry, if for every pair of points u,v 2 M we have:

• d(u,v) = d’((u),(v)).

• Clearly, isometric spaces are indistingushable as far as metric properties are concerned.

Similarity I

• Let (M,d) and (M’,d’) be two metric spaces. A mapping h:M ! M’ with the property that for any four points a,b,c,d 2 M we have:

• If d(a,b) = d(c,d) then d(h(a),h(b)) = d(h(c),h(d)) is called similarity (of type I).

Similarity II

• Let (M,d) and (M’,d’) be two metric spaces and r 2 R\{0}. A mapping h:M ! M’ with the property that for any pair of points a,b, 2 M we have:

• If d(a,b) = r d(h(a),h(b)) then h is called similarity (of type II) and r is called the dilation factor.

Type I vs. Type II

• Clearly each similarity of type II is also a similarity of type I. In general, the converse is false.

• Theorem. A similarity on (Rn,d2) of type I is also of type II. (Proof can be found in Paul B. Yale: Geomerty and Symmetry, Dover, 1988 (reprint from 1968))

Finite Metric Space

• In a finite metric space (M,d) we may assume that min d(u,v) = 1. Max d(u,v) is called the diameter of M. The quotient Max d(u,v)/Min d(u,v) is called dilation coefficient.

Three Classical Results

• The Steinitz Theorem, Fary’s Theorem and Tutte’s Theorem can be interpreted as graph representations.

The Energy

• Usually we try to find among the representations of a certain type the one that is “optimal” in certain sense.

• To this end we may define an energy function E() and then seek a representation that minimizes the energy.

• There are several such energy functions used in various problem areas.

Some Energy Models

• Spring embedders

• Molecular mechanics

• Tutte drawing

• Schlegel diagram (B. Plestenjak).

• [Connection to Markov Chains]

• ...

• Laplace Representation

The Laplace Representation• Let be a representation in Rk. Define

E() = uv 2 E(G) ||(u)-(v)||2

• It turns out that the minimum (under some reasonable conditions) is achieved as follows.

1. Take the Laplace matrix of G. 2. Q(G) = D(G)-A(G)3. Find the eigenvalues 0 = 1 · 2 · ... · n.4. Find the corresponding orthonormal eigenvectors x1, x2,

..., xn.5. Form a matrix R =[x2|x3| ... |xk+1]6. Let (vi) = rowi(R).

An R3 Laplace representation of a fullerene (skeleton of a trivalent polyhedron with pentagonal and

hexagonal faces)

Nodal Domains - Revisited

• The Example on the left represents nodal domains obtained from the Laplace representation of G(10,4).

Congruence and Similarity

• A representation in any metric sapce, in particular in Rn, can be scaled without “being changed too much”. If is injective on the vertices, we may scale it in such a way that Min d(u,v) = 1, for all u ~ v. Each vertex faithful representation is similar to a standard one.

Similar Representations

• Let :G ! M be graph representations into a metric space M. We say they are similar, if there exists a similarity h 2 Sim(M) such that for each v 2 V(G) we have (v) := h((v)).

• We would like to assign the same energy to similar representions.

Unit Distance Graphs

• Let be a representation in Rk. Define

Ep () = (uv 2 E(G) ||(u)-(v)||p) (1/p)

• We assume that Min uv 2 E(G) ||(u)-(v)|| = 1

• In the limit when p ! 1 we get

E1 () = Maxuv 2 E(G) ||(u)-(v)||

• The number E1 () is called dilation coefficient. • Hence E1 () ¸ 1. In the special case: E1 () = 1 we

call this representation a unit distance graph.

Generalized Petersen Graphs

• Some generalized Petersen graphs admit unit distance representations in R2 with considerable symmetry.

• Several questions:• Which G(n,k) have unit

distance representations?

• Connection to symmetry.

• Change dimension.

Symmetry of Representation

• Let :G ! M be a graph representation into a metric space M. Let Aut be the group of symmetries of this representation. Namely g 2 Aut G is a symmetry of (and therefore g 2 Aut ) if there exists an isometry h 2 Iso(M) such that for each v 2 V(G) we have (g(v)) = h((v)) and for each e=uv 2 E(G) we have d((u),(v)) = d((g(u)),(g(v)).

Representations with Symmetry(Motivation: Recent work on regular polygons and

regular polyhedra by Branko Grünbaum)

• Let G be a graph and let Aut(G) be its automorphism group. • Let Iso(Rk) be the group of Euclidean isometries.• We say that an automorphism a 2 Aut(G) is preserved by the

representation if there exists an isometry 2 Iso(Rk) such that

• for each vertex v 2 V(G) it follows that ((v)) = (a(v)).• The set of all automorhpisms 2 Aut(G) that are preseved

by forms a group that we call the symmetry group of the representation .

• A representation with trivial symmetry group is called rigid.

An Example• Consider the one-dimensional

representation of the triangle C3 with V(C3) = {1,2,3}.

• Aut(C3) = S3 = {id,(12),(13),(23),(123),(132)}.

• Let ri = (i). W.l.o.g. assume r3 = 0. Hence each representation can be viewed as a point in the (r1,r2) – plane.

• The points not lying on any of the axes or lines determine a rigid representation. Each line is labeled by its symmetry group. The origin retains the whole symmetry.

• Note that the underlined representations are non-singular (meaning that is one-to-one)..

(12)

(13)

(12)(12)

(13)(13)

(23)(23)

(23)

1 2

3

r1 0 =r3r2

A General Problem

• For an arbitray graph G find a non-singular representation in R2 minimizing the number of vertex orbits or edge orbits.

• There are several obvious variations to this problem.

Homework 7• H1. It is easy to verify that K4 is

not a unit distance graph in the plane. Consider a drawing of K4 in the plane with only two distinct edge lengths. How many such non-isomorphic drawings are there? (Hint: there are six). Compute the dilation coefficient for all such drawings.

Chapter 2. Statistics Page

• Number of slides:282

• Number of sections:7

• Number of exercises:32

• Number of homeworks:21