Symmetry in Graphs

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)
- a (1)(3)(2 4)
- b (1 3)(2)(4)
- g a b (1 3)(2 4)

a

1

2

c

b

d

e

3

4

Example for Aut G acting on E(G).

- Aut G 4.
- EG a,b,c,d,e
- Id (a)(b)(c)(d)(e)
- a (a d)(b c)(e)
- b (a b)(c d)(e)
- g a b (a c)(b d)(e)

a

1

2

c

b

d

e

3

4

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 a(1) 1, a(2)

4, we have a(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 S2,3. - E(G) a,b,c,d,e has two orbits Z a,b,e,d

and Mc.

a

1

2

c

b

d

e

3

4

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.

a(v)

b

a

v

ba(v)

LEFT

a(v)

b

a

v

ab(v)

RIGHT

Cayley Color Digraph for D3.

X

- Right Cayley Color Digraph
- Convention Since 1 Y2 we may use the

undirected version of the edge..

XY

Y

X2Y

1

X2

X

Y

Cayley Graph (Right)

- Let G be a group and W ½ Ga set of generators,

such that - Symmetric W W-1
- Does not contain identity 1 Ï W.
- To a pair (G,W) we can associate a Cayley graph X

Cay(G,W) as follows - V(X) G
- g h , g-1h 2 W.

Basic Theorem about Cayley graphs

- Graph X is a Cayley graph, if and only if there

exists a subgroup G Aut X, acting regularly on

V(X)! - Exercise Prove that 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 Free product of groups and

explore the corresponding product construction of

rooted Cayley graphs.

Frucht Theorem

- Theorem For each finite group G there exists a

graph X, such that G isomorphic to Aut X.

Vertex-Transitive Graphs

- If group G acts on a space V with a single orbit

(x V), we say that the action is transitive. - Let (G,V) be a permutation group and let x be

any of its orbits. Restriction (G,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.

Exercises

- N1 Prove that G(n,k) is vertex transitive, if

and only if k2 1 mod n, or else n10 and

k2. - N2 Prove that Cn, Kn, Qn are all vertex

transitive. - N3 Which complete multipartite graphs Ka,b,

Ka,b,c, ... are vertex transitive? - N4 Prove that the Cartesian product of vertex

transitive graphs is vertex transitive.

Vertex-Transitive Subgraphs

- Let G be a graph and x ½ V(G) and orbit for Aut

G. The induced subgraph ltxgt is vertex

transitive. - Let H ½ G be an induced subgraph of G. Let G lt

Aut H be the group of those automorphisms that

can be extended to the group of automorphisms of

G. - Given H and given G lt 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, Möbius

ladder M4 and prism P6. Which graphs are edge

transitive?

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.

Edge Transitive Graphs that are not Vertex

Transitive

- Theorem An edge transitive graph X, that is not

vertex transitive is bipartite. - Lemma If both endvertices of an edge 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, is 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 and automorphism a 2 Aut

X, mapping e to f. a(e ) f. Therefore it

maps e- to f-. a(e- ) f- and furthermore

a(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 a 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).

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 aiai1 is an edge

in E(X) and ai-1 ? ai1. - 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

(s1)-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. - 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.

-1

-4

4

1

2

-2

Z9

Half Arc Transitive Graph

- There are several families of ½-arc-transitive

graph (many discovered by mathematicians in

Slovenia). - Theorem Each ½-arc-transitive graph is regular,

of even valence. - 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) gt outdeg(v), the same inequality would

hold at every vertex. This contradicts the

well-known fact - S indeg(x) S outdeg(x).

LCF Notation for Cubic Graphs

- Cubic graph X on 2n vertices, with a given

Hamilton cycle, can be easily encoded by

successive lengths of the cords along the

Hamiltono cycle. - Example Graph on the left
- LCF3,4,2,3,4,2 LCF3,-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 denote the Heawood graph.

Exercises

- N1 Write a LCF code for the Dürer graph.
- N2 Write a LCF code for K4.
- N3 Write a LCF code for M3 K3,3. Generalize to

Möbius ladder Mn.

Edge Orbits of Vertex Transitivne 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 f e. It follows from vertex

transitivity. 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 Vertex-transitive graph X, such that

Aut X V(X) is called a graphical regular

representation (GRR) of group G Aut X. - Remark If Aut X acts transitively on V(X), it

does not mean that there exists a subgroup G

Aut X, actinng on V(X) regularly.

0-Symmetric Graphs

- Definition 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.

Stabilizer ?x of vertex x acts on the set of

neighbors of x X(x). - Lemma In a vertex transitive graph, the number w

edge orbits equals to the number of orbits when

?x acts 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(9S) H(28 27 25),

where S 0, 1, 3. - LCF5,-59.

The Mark Watkins Graph

- Smallest 0-symmetric Haar graph H(n0,a,b) with

the property gcd(a,n) gt 1, gcd(b,n) gt

1,gcd(b-a,n) gt 1, gcd(a,b) 1 has parameters n

30, a 2, b 5. It is called the Mark Watkins

graph.

Semi Symmetric Graphs.

- Definition Regular graph X, that is edge

transitive, but not vertex transitive, is called

semisymmetric. - On the left we see one of them, the 4 valent

Folkman graph.

Direct Product of Groups - Revisited.

- A B direct product of groups defined on the

cartesian product. 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.

Exercises

- N1 Prove that Z3 Z3 À Z9.
- N2 Prove that Z2 Z3 _at_ Z6.
- N3() Prove that any finite abelian group A is

isomorphic to the direct product A(n1,n2,...,nk)

Zn1 Zn ... Znk, where n1n2...nk. - N4() Prove that the groups A(n1,n2,...,nk)

A(m1,m2,...,mj). with n1n2...nk and

m1m2...mj are equal if and only if j k and

ntmt, for each t.

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 Md(a,x) r Ball centered in

a with radius r. - Let S(a,r) x 2 Ms(a,x) r Sphere centered

in 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. It 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, 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. - Exercise. Find an isotropic metric space that is

not homogeneous. - Let X ½ M.
- Iso(M,X) is the group of isometries fixing X

set-wise. - Iso(Mrel 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 - 4, nonbipartite, grith 3, K4
- 6, bipartite, girth 4, K3,3
- 10, nonbipartite, girth 5, G(5,2)
- 8, bipartite, girth 4, Q3
- 14, bipartite, girth 6, Heawood
- 18, bipartite, girth 6, Pappus
- 28, nonbipartite, girth 7, Coxeter
- 30, bipartite, grith 8, Tutte 8-cage

Cubic Distance Transitive Graphs

- Theorem There are only 12 cubic distance

transitive graphs - 09. 20, nonbipartite, grith 5, G(10,2)
- 10. 30, bipartite, girth 6, G(10,3)
- 11. 102, nonbipartite, girth 9, Biggs Smith

H(171,2,4,8) - 12. 90, bipartite, grith 10,Foster

Example Foster Graph

- The bipartite Foster graph on 90 vertices is

largest cubic distance transitive graph. - LCF17,-9,37,-15

Biggs-Smith Graph

- Biggs-Smith graph H(171,2,4,8) has 102 vertices

and girth 9.

Biggs-Smith Graph

- Biggs-Smith graph H(171,2,4,8) has 102 vertices

and girth 9. - Its Kronecker cover is bipartite nad 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.

Homework

- H1. Find a better drawing of Gewirtz graph.

Quartic Distance Transitive Graphs

- Theorem There are only 15 quartic distance

transitive graphs - K5
- K4,4
- L(K4)
- L(K3,3)
- L(G(5,2))

Quartic Distance Transitive Graphs

- L(Heawood)
- K2 K5
- Heawood3.
- (4,6) cage
- Gewirtz graph O4.

Quartic Distance Transitive Graphs

- L(Tutte8cage)
- Q4
- 4-fold cover of K4,4
- (4,12) cage
- K2 O4.

Homework

- 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 for

a Cayley graph.

Hamiltonicity

- Most vertex-transitive graphs have Hamilton

cycles. - There are only 4 known graphs without Hamilton

cycle. All four of them have Hamilton path.

Similar Representations

- Let r,sG ! 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 s(v) h(r(v)). - We would like to assign the same energy to

similar representions.

Symmetry of Representation

- Let rG ! M be a graph representation into a

metric space M. Let Aut r be the group of

symmetries of this representation. Namely g 2

Aut G is a symmetry of r (and therefore g 2 Aut

r) if there exists an isometry h 2 Iso(M) such

that for each v 2 V(G) we have r(g(v)) h(r(v))

and for each euv 2 E(G) we have d(r(u),r(v))

d(r(g(u)),r(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 representation r if there exists an

isometry a 2 Iso(Rk) such that - for each vertex v 2 V(G) it follows that a(r(v))

r(a(v)). - The set of all automorhpisms Gr 2 Aut(G) that are

preseved by r forms a group that we call the

symmetry group of representation r. - Representation with a trivial symmetry group is

called rigid.

An Example

(13)

- Consider onedimensional representation of the

triangle C3 with V(C3) 1,2,3. - Aut(C3) S3 id,(12),(13),(23),(123),(132).
- Let ri r(i). Without loss of generality 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 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 r is one-to-one)..

(23)

(12)

(13)

(23)

(12)

3

r1

r2

0 r3

1

2

A 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.