Symmetric graphs of valency five and their basic normal quotients

Abstract

Abstract A graph Γ is symmetric or arc-transitive if its automorphism group Aut ( Γ ) is transitive on the arc set of the graph, and Γ is basic if Aut ( Γ ) has no non-trivial normal subgroup N such that the quotient graph Γ N has the same valency as Γ . In this paper, we classify symmetric basic graphs of order 2 q p n and valency 5, where q p are two primes and n is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order 2 q with 5 | ( q − 1 ) , the complete graph K 6 of order 6, the complete bipartite graph K 5 , 5 of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order k p n for some small integers k and n are classified.