2-Arc-transitive Cayley graphs on alternating groups

Abstract

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime valency (provided solvable vertex stabilizers) are Cayley graphs on alternating groups An. This naturally motivates the study of 2-arc-transitive Cayley graphs on An 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 An with G almost simple, the socle of G is either An+1 or An+2. We also construct the first infinite family of (An+2,2)-arc-transitive Cayley graphs on An.