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

TL;DR

This paper classifies certain highly symmetric Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven, showing they are either normal or have specific arc-transitivity properties, with explicit examples constructed.

Contribution

It provides a complete classification of arc-transitive Cayley graphs with valency seven on non-abelian simple groups with soluble stabilizers, identifying conditions for normality and existence of specific examples.

Findings

Either the graph is a normal Cayley graph or it is S-arc-transitive with specific group pairs.

Explicit examples of 7-valent non-normal Cayley graphs on alternating groups are constructed.

Identifies the exact parameters (n=7,21,63,84) where such graphs exist.

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.