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)• = (1)(3)(2 4)• = (1 3)(2)(4)• = = (1 3)(2 4)

1

3 4

2a

e

c bd

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)

1

3 4

2a

e

c bd

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}.

1

3 4

2a

e

c bd

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..

1

X2

X

Y

XY

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 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 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) be a permutation group and let [x] be any of its orbits. 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.

Exercises

• N1: Prove that G(n,k) is vertex transitive, if and only if k2 § 1 mod n, or else n=10 and k=2.

• 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 <[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, Möbius ladder M4 and prism 6. 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 2 Aut X, mapping e+ to f+. (e+ ) = f+. Therefore it maps e- to f-. (e- ) = f- and furthermore (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).

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.

• 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

-1+4

-2

+2

-4

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) > 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

• 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:

• 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] 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 = Aut X.

• Remark: If Aut X acts transitively on V(X), it does not mean that there exists a subgroup · 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 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(9;S) = H(28 + 27 + 25), where S = {0, 1, 3}.

• LCF[{5,-5}9].

The Mark Watkins Graph

• Smallest 0-symmetric Haar graph H(n;{0,a,b}) with the property gcd(a,n) > 1, gcd(b,n) > 1,gcd(b-a,n) > 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 Z6.

• N3(*): Prove that any finite abelian group A is isomorphic to the direct product A(n1,n2,...,nk) = Zn1

£ Zn ... £ Znk, where n1|n2|...|nk.

• N4(*): Prove that the groups A(n1,n2,...,nk) = 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.

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} Ball centered in a with

radius r.• Let S(a,r) = {x 2 M|s(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(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, grith = 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, grith = 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 largest cubic distance transitive graph.

• LCF[{17,-9,37},-15]

Biggs-Smith Graph

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

Biggs-Smith Graph

• Biggs-Smith graph H(17;1,2,4,8) 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:

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.

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 :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.

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

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 representation .

• Representation with a trivial symmetry group is called rigid.

An Example• Consider onedimensional representation of

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

• Aut(C3) = S3 = {id,(12),(13),(23),

(123),(132)}.• Let ri = (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 is one-to-one)..

(12)

(13)

(12)(12)

(13)(13)

(23)(23)

(23)

1 2

3

r1 0 =r3r2

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.