Arc-transitive Cayley graphs on non-abelian simple groups with soluble vertex stabilizers and valency seven

Abstract

In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let $\\Ga$ be such a Cayley graph on a non-abelian simple group $T$. It is proved that either $\\Ga$ is a normal Cayley graph or $\\Ga$ is $S$-arc-transitive, with $(S,T)=(\\A_n,\\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on $\\A_{n-1}$ and specific examples are constructed.