Arc-transitive and s-regular Cayley graphs of valency 5 on Abelian groups

Abstract

Let G be a flnite group, and let 1G 62 S µ G. A Cayley di-graph i = Cay(G;S) of G relative to S is a di-graph with a vertex set G such that, for x;y 2 G, the pair (x;y) is an arc if and only if yx i1 2 S. Further, if S = S i1 := fs i1 js 2 Sg, then i is undirected. i is conected if and only if G = hsi. A Cayley (di)graph i = Cay(G;S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of i. A graph i is said to be arc-transitive, if Aut(i) is transitive on an arc set. Also, a graph i is s-regular if Aut(i) acts regularly on the set of s-arcs. In this paper, we flrst give a complete classiflcation for arc-transitive Cayley graphs of valency flve on flnite Abelian groups. Moreover, we classify s-regular Cayley graph with valency flve on an abelian group for each s ‚ 1.