Symmetric graphs of prime valency with a transitive simple group

TL;DR

This paper investigates symmetric Cayley graphs of prime valency, identifying exceptions where such graphs are not normal, by analyzing maximal factorizations of finite almost simple groups.

Contribution

It provides a classification of exceptions for symmetric Cayley graphs of prime valency using group factorization techniques.

Findings

Identifies potential exceptions to normality in symmetric Cayley graphs of prime valency.

Uses maximal factorizations of finite almost simple groups to determine exceptions.

Extends previous results by Fang et al. on normality of such graphs.

Abstract

A graph $\Ga=(V,E)$ is called a Cayley graph of some group $T$ if the automorphism group $\Aut(\Ga)$ contains a subgroup $T$ which acts on regularly on $V$. If the subgroup $T$ is normal in $\Aut(\Ga)$ then $\Ga$ is called a normal Cayley graph of $T$. Let $r$ be an odd prime. Fang et al. \cite{FMW} proved that, with a finite number of exceptions for finite simple group $T$, every connected symmetric Cayley graph of $T$ of valency $r$ is normal. In this paper, employing maximal factorizations of finite almost simple groups, we work out a possible list of those exceptions for $T$.