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

TL;DR

This paper classifies half-arc-transitive graphs of order p^3 with valencies 6 and 8, including the first known infinite family on non-metacyclic p-groups, expanding understanding of such graphs.

Contribution

It provides a classification of half-arc-transitive graphs of order p^3 with valencies 6 and 8, introducing the first infinite family on non-metacyclic p-groups.

Findings

Classified half-arc-transitive graphs of order p^3 and valency 6 or 8.

Constructed the first infinite family of such graphs on non-metacyclic p-groups.

Extended known classifications beyond valency 4.

Abstract

A graph is called {\em 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 tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] 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.