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

TL;DR

This paper classifies the structure of automorphism groups of connected pentavalent symmetric graphs with non-abelian simple groups acting transitively, identifying possible group pairs and improving previous classifications.

Contribution

It provides a detailed classification of automorphism groups for these graphs, identifying 58 possible group pairs and refining earlier results on pentavalent symmetric Cayley graphs.

Findings

58 possible pairs of non-abelian simple groups identified

17 pairs when G is arc-transitive

13 pairs when G is regular on vertices

Abstract

A graph \Gamma is said to be {\em symmetric} if its automorphism group \Aut(\Gamma) is transitive on the arc set of \Gamma. Let $G$ be a finite non-abelian simple group and let \Gamma be a connected pentavalent symmetric graph such that G\leq \Aut(\Gamma). In this paper, we show that if $G$ is transitive on the vertex set of \Gamma, then either G\unlhd \Aut(\Gamma) or \Aut(\Gamma) contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is one of $58$ possible pairs of non-abelian simple groups. In particular, if $G$ is arc-transitive, then $(G,T)$ is one of $17$ possible pairs, and if $G$ is regular on the vertex set of \Gamma, then $(G,T)$ is one of $13$ possible pairs, which improves the result on pentavalent symmetric Cayley graph given by Fang, Ma and Wang in 2011.