Vertex‐transitive graphs: Symmetric graphs of prime valency

Abstract

Let G be a group acting symmetrically on a graph 2, let G, be a subgroup of G minimal among those that act symmetrically on 8, and let G2 be a subgroup of G, maximal among those normal subgroups of GI which contain no member except 1 which fixes a vertex of Z. The most precise result of this paper is that if Z has prime valency p , then either Z is a bipartite graph or G2 acts regularly on Z or GI I G2 is a simple group which acts symmetrically on a graph of valency p which can be constructed from C and does not have more vertices than 2. The results on vertextransitive groups necessary to establish results like this are also included.

Pentavalent symmetric graphs of order 16p

“…In view of [24,Theorem 9], we have the following: Proposition 2.4 Let X be a connected pentavalent (G, s)-arc-transitive graph for some s ≥ 1, and let N be a normal subgroup of G with more than two orbits on V (X). Then X N is also a pentavalent symmetric graph and N is the kernel of the action of G on V (X N ).…”

“…Furthermore, g can be chosen as a 2-element in A, and the valency of X is |D|/|H| = |H : H ∩ H g |. For more details regarding coset graphs, see, for example, [9,24,33]. Now we introduce three pentavalent symmetric graphs of order 6p for some prime p which were constructed in [17,Section 3].…”

Pentavalent symmetric graphs of order 2p 3

“…Furthermore, g can be chosen as a 2-element in A, and the valency of X is |D|/|H| = |H : H ∩ H g |. For more details regarding coset graphs, see, for example, [9,24,33]. Now we introduce three pentavalent symmetric graphs of order 6p for some prime p which were constructed in [17, Section 3].…”

Involution Graphs Where the Product of Two Adjacent Vertices Has Order Three

“…To obtain a necessary and sufficient condition for an S 3 -involution graph (G, X, S) to be connected with X a single conjugacy class, we need to introduce a general method for constructing vertex-transitive graphs (see for example [20]). …”