2-Arc-transitive 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 ${\mathrm{A}}_n$.