2-Arc-transitive Cayley graphs on alternating groups

TL;DR

This paper investigates 2-arc-transitive Cayley graphs on alternating groups, characterizing their automorphism groups and constructing an infinite family of such graphs with specific properties.

Contribution

It provides a complete characterization of automorphism groups of 2-arc-transitive Cayley graphs on alternating groups and constructs the first infinite family of such graphs.

Findings

Automorphism groups have socle either A_{n+1} or A_{n+2}.

Constructed the first infinite family of (A_{n+2},2)-arc-transitive Cayley graphs on A_n.

Extended understanding of symmetry properties of Cayley graphs on alternating groups.

Abstract

An interesting fact is that most of the known connected $2$-arc-transitive nonnormal Cayley graphs of small valency on finite simple groups are $(\mathrm{A}_{n+1},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$. This motivates the study of $2$-arc-transitive Cayley graphs on $\mathrm{A}_n$ for arbitrary valency. In this paper, we characterize the automorphism groups of such graphs. In particular, we show that for a non-complete $(G,2)$-arc-transitive Cayley graph on $\mathrm{A}_n$ with $G$ almost simple, the socle of $G$ is either $\mathrm{A}_{n+1}$ or $\mathrm{A}_{n+2}$. We also construct the first infinite family of $(\mathrm{A}_{n+2},2)$-arc-transitive Cayley graphs on $\mathrm{A}_n$.