Pentavalent symmetric graphs admitting vertex-transitive non-abelian simple groups

Abstract

A graph Γ is said to be symmetric if its automorphism group Aut(Γ) is transitive on the arc set of Γ. Let G be a finite non-abelian simple group and let Γ be a connected pentavalent symmetric graph with G≤Aut(Γ). In this paper, we show that if G is transitive on the vertex set of Γ, then either G⊴Aut(Γ) or Aut(Γ) contains a non-abelian simple normal subgroup T such that G≤T and (G,T) is one of 58 possible pairs of non-abelian simple groups. In particular, if G is transitive on the arc set of Γ, then (G,T) is one of 17 possible pairs, and if G is regular on the vertex set of Γ, then (G,T) is one of 13 possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, et al. (2011).