6-Valent Arc-Transitive Cayley Graphs on Abelian Groups

Let G be a finite group and S be a subset of G such that 1G ̸∈ S and S −1 = S. The Cayley graph Σ = Cay(G, S) on G with respect to S is the graph with the vertex set G such that, for §, † ∈ G, the pair (§, †) is an arc in Cay(G, S) if and only if †§−1 ∈ S. The graph Σ is said to be arc-transitive if its full automorphism group Aut(Σ) is transitive on its arc set. In this paper we give a classification for arc-transitive Cayley graphs with valency six on finite abelian groups which are non-normal. Moreover, we classify all normal Cayley graphs on non-cyclic abelian groups with valency 6.