Symmetry in Graphs

1
Symmetry in Graphs
2
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)).

3
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
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
5
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.

6
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
7
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
8
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
9
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
10
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.

11
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.

12
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.

13
Frucht Theorem

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

14
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.

15
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?

16
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.

17
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.

18
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.

19
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?

20
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.

21
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.

22
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.

23
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.

24
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.

25
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).

26
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).

27
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).

28
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.

29
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.

30
1/2-Arc-Transitive Graphs

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

31
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.

32
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
33
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).

34
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

35
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.

36
Heawood Graph - LCF

• LCF(5)-7 denote the Heawood graph.

37
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.

38
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.

39
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.

40
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.

41
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.

42
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.

43
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.

44
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.

45
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.

46
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.

47
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.

48
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.

49
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.

50
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.

51
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.

52
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.

53
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.

54
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

55
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

56
Example Foster Graph

• The bipartite Foster graph on 90 vertices is
  largest cubic distance transitive graph.
• LCF17,-9,37,-15

57
Biggs-Smith Graph

• Biggs-Smith graph H(171,2,4,8) has 102 vertices
  and girth 9.

58
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.

59
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.

60
Homework

• H1. Find a better drawing of Gewirtz graph.

61
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))

62
Quartic Distance Transitive Graphs

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

63
Quartic Distance Transitive Graphs

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

64
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.

65
Hamiltonicity

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

66
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.

67
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)).

68
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.

69
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
70
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.