Symmetric graphs with 2-arc transitive quotients Guangjun Xu and - PowerPoint PPT Presentation

Symmetric graphs with 2-arc transitive quotients Guangjun Xu and Sanming Zhou Department of Mathematics and Statistics The University of Melbourne Australia Shanghai Jiao Tong University, 20/7/2013

automorphism group ◮ Let Γ be a graph.

automorphism group I Let Γ be a graph. I An automorphism of Γ is a permutation of the vertex set which preserves adjacency and nonadjacency relations.

automorphism group I Let Γ be a graph. I An automorphism of Γ is a permutation of the vertex set which preserves adjacency and nonadjacency relations. I The group Aut (Γ) = { automorphisms of Γ } under the usual composition of permutations is called the automorphism group of Γ.

arcs and s -arcs I An arc is an oriented edge.

arcs and s -arcs I An arc is an oriented edge. I One edge { α, β } gives rise to two arcs ( α, β ), ( β, α ).

arcs and s -arcs I An arc is an oriented edge. I One edge { α, β } gives rise to two arcs ( α, β ), ( β, α ). I An s -arc is a sequence α 0 , α 1 , . . . , α s of s + 1 vertices such that α i , α i +1 are adjacent and α i − 1 � = α i +1 .

arcs and s -arcs I An arc is an oriented edge. I One edge { α, β } gives rise to two arcs ( α, β ), ( β, α ). I An s -arc is a sequence α 0 , α 1 , . . . , α s of s + 1 vertices such that α i , α i +1 are adjacent and α i − 1 � = α i +1 . I An oriented path of length s is an s -arc, but the converse is not true.

symmetric and highly arc-transitive graphs I Let G ≤ Aut (Γ).

symmetric and highly arc-transitive graphs I Let G ≤ Aut (Γ). I Γ is G -vertex transitive if G is transitive on V (Γ).

symmetric and highly arc-transitive graphs I Let G ≤ Aut (Γ). I Γ is G -vertex transitive if G is transitive on V (Γ). I Γ is G -symmetric if it is G -vertex transitive and G is transitive on the set of arcs of Γ.

symmetric and highly arc-transitive graphs I Let G ≤ Aut (Γ). I Γ is G -vertex transitive if G is transitive on V (Γ). I Γ is G -symmetric if it is G -vertex transitive and G is transitive on the set of arcs of Γ. I Γ is ( G , s )-arc transitive if it is G -vertex transitive and G is transitive on the set of s -arcs of Γ.

symmetric and highly arc-transitive graphs I Let G ≤ Aut (Γ). I Γ is G -vertex transitive if G is transitive on V (Γ). I Γ is G -symmetric if it is G -vertex transitive and G is transitive on the set of arcs of Γ. I Γ is ( G , s )-arc transitive if it is G -vertex transitive and G is transitive on the set of s -arcs of Γ. I ( G , s )-arc transitivity ⇒ ( G , s − 1)-arc transitivity ⇒ · · · ⇒ ( G , 1)-arc transitivity (= G -symmetry)

two observations I Let G α := { g ∈ G : g fixes α } be the stabiliser of α ∈ V (Γ) in G .

two observations I Let G α := { g ∈ G : g fixes α } be the stabiliser of α ∈ V (Γ) in G . I Γ is G -symmetric ⇔ G is transitive on V (Γ) and G α is transitive on Γ( α ) (neighbourhood of α in Γ).

two observations I Let G α := { g ∈ G : g fixes α } be the stabiliser of α ∈ V (Γ) in G . I Γ is G -symmetric ⇔ G is transitive on V (Γ) and G α is transitive on Γ( α ) (neighbourhood of α in Γ). I Γ is ( G , 2)-arc transitive ⇔ G is transitive on V (Γ) and G α is 2-transitive on Γ( α ).

two observations I Let G α := { g ∈ G : g fixes α } be the stabiliser of α ∈ V (Γ) in G . I Γ is G -symmetric ⇔ G is transitive on V (Γ) and G α is transitive on Γ( α ) (neighbourhood of α in Γ). I Γ is ( G , 2)-arc transitive ⇔ G is transitive on V (Γ) and G α is 2-transitive on Γ( α ). I The analogy is not true when s ≥ 3.

examples I The dodecahedron graph is A 5 -arc transitive.

examples I The dodecahedron graph is A 5 -arc transitive. I For n ≥ 4, K n is 2-arc transitive but not 3-arc transitive.

examples I The dodecahedron graph is A 5 -arc transitive. I For n ≥ 4, K n is 2-arc transitive but not 3-arc transitive. I For n ≥ 3, K n , n is 3-arc transitive but not 4-arc transitive.

examples Tutte’s 8-cage is 5-arc transitive. It is a cubic graph of girth 8 with minimum order (30 vertices).

motivation I A G -symmetric graph Γ, which is not necessarily ( G , 2)-arc transitive, may admit a natural ( G , 2)-arc transitive quotient with respect to a G -invariant partition.

motivation I A G -symmetric graph Γ, which is not necessarily ( G , 2)-arc transitive, may admit a natural ( G , 2)-arc transitive quotient with respect to a G -invariant partition. I When does this happen? (Iranmanesh, Praeger and Z, 2005)

motivation I A G -symmetric graph Γ, which is not necessarily ( G , 2)-arc transitive, may admit a natural ( G , 2)-arc transitive quotient with respect to a G -invariant partition. I When does this happen? (Iranmanesh, Praeger and Z, 2005) I If there is such a quotient, what information does it give us about the original graph? (Iranmanesh, Praeger and Z, 2005) Observation If Γ admits a ( G , 2)-arc transitive quotient, then a natural 2-point transitive and block transitive design D ∗ ( B ) arises and plays a significant role in understanding the structure of Γ.

known results I Li, Praeger and Zhou (2000): k = v − 1

known results I Li, Praeger and Zhou (2000): k = v − 1 I Iranmanesh, Praeger and Zhou (2005): k = v − 2

known results I Li, Praeger and Zhou (2000): k = v − 1 I Iranmanesh, Praeger and Zhou (2005): k = v − 2 I Li, Praeger and Zhou (2010): k = v − 2 and a natural auxiliary graph is a cycle

known results I Li, Praeger and Zhou (2000): k = v − 1 I Iranmanesh, Praeger and Zhou (2005): k = v − 2 I Li, Praeger and Zhou (2010): k = v − 2 and a natural auxiliary graph is a cycle I Lu and Zhou (2007): constructions were given when D ∗ ( B ) or its complement is degenerate

this talk I We give necessary conditions for a natural quotient of Γ to be ( G , 2)-arc transitive when v − k is a prime.

this talk I We give necessary conditions for a natural quotient of Γ to be ( G , 2)-arc transitive when v − k is a prime. I When v − k = 3 or 5, these necessary conditions are essentially sufficient.

notation I Γ: G -symmetric graph

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b I k = | B ∩ Γ( C ) | : where B , C ∈ B are adjacent blocks, and Γ( C ) is the neighbourhood of C in Γ

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b I k = | B ∩ Γ( C ) | : where B , C ∈ B are adjacent blocks, and Γ( C ) is the neighbourhood of C in Γ I r : number of blocks containing neighbours of a fixed vertex

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b I k = | B ∩ Γ( C ) | : where B , C ∈ B are adjacent blocks, and Γ( C ) is the neighbourhood of C in Γ I r : number of blocks containing neighbours of a fixed vertex I Γ B ( B ): neighbourhood of B in Γ B

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b I k = | B ∩ Γ( C ) | : where B , C ∈ B are adjacent blocks, and Γ( C ) is the neighbourhood of C in Γ I r : number of blocks containing neighbours of a fixed vertex I Γ B ( B ): neighbourhood of B in Γ B I D ( B ): incidence structure with point set B and block set Γ B ( B ), in which α ∈ B and C ∈ Γ B ( B ) are incident if and only if α ∈ Γ( C )

notation I Γ: G -symmetric graph I B : nontrivial G -invariant partition of V (Γ) with block size v I Γ B : quotient with respect to B , with valency b I k = | B ∩ Γ( C ) | : where B , C ∈ B are adjacent blocks, and Γ( C ) is the neighbourhood of C in Γ I r : number of blocks containing neighbours of a fixed vertex I Γ B ( B ): neighbourhood of B in Γ B I D ( B ): incidence structure with point set B and block set Γ B ( B ), in which α ∈ B and C ∈ Γ B ( B ) are incident if and only if α ∈ Γ( C ) I D ( B ) is a 1-( v , k , r ) design with b blocks (Gardiner and Praeger 1995)

2-arc transitive quotients I We always assume Γ B is connected.

2-arc transitive quotients I We always assume Γ B is connected. I Γ B is always G -symmetric, and sometimes ( G , 2)-arc transitive (even if Γ is not).

2-arc transitive quotients I We always assume Γ B is connected. I Γ B is always G -symmetric, and sometimes ( G , 2)-arc transitive (even if Γ is not). I D ∗ ( B ): dual of D ( B ) (swap ‘points’ and ‘blocks’)

2-arc transitive quotients I We always assume Γ B is connected. I Γ B is always G -symmetric, and sometimes ( G , 2)-arc transitive (even if Γ is not). I D ∗ ( B ): dual of D ( B ) (swap ‘points’ and ‘blocks’) I D ∗ ( B ): complementary of D ∗ ( B ) (swap ‘flags’ and ‘antiflags’)