Half-arc-transitive graphs of prime-cube order of small valencies

Abstract

A graph is called half-arc-transitive if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime p there is no half-arc-transitive graph of order p or p 2 . In 1992, Xu classified half-arc-transitive graphs of order p 3 and valency 4 . In this paper we classify half-arc-transitive graphs of order p 3 and valency 6 or 8 . In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic p -groups is constructed.