On cubic s-arc transitive Cayley graphs of finite simple groups

Abstract

For a positive integer s, a graph Γ is called s- arc transitive if its full automorphism group Aut Γ acts transitively on the set of s-arcs of Γ. Given a group G and a subset S of G with S= S −1 and 1∉ S, let Γ=Cay( G, S) be the Cayley graph of G with respect to S and G R the set of right translations of G on G. Then G R forms a regular subgroup of Aut Γ. A Cayley graph Γ=Cay( G, S) is called normal if G R is normal in Aut Γ. In this paper we investigate connected cubic s-arc transitive Cayley graphs Γ of finite non-Abelian simple groups. Based on Li’s work (Ph.D. Thesis (1996)), we prove that either Γ is normal with s≤2 or G= A 47 with s=5 and Aut Γ ≅ A 48. Further, a connected 5-arc transitive cubic Cayley graph of A 47 is constructed.