On the Semisymmetric Graphs of Order \(2p^3\): Faithful and Primitive Case

Abstract

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. A semisymmetric graph must be bipartite whose automorphism group has two orbits of the same size on the vertices. One of our long-term goals is to determine all the semisymmetric graphs of order \(2p^3\), for any prime \(p\). All these graphs \(\Gamma\) with the automorphism group \(Aut(\Gamma)\), are divided into two subclasses: (I) \(Aut(\Gamma)\) acts unfaithfully on at least one bipart; and (II) \(Aut(\Gamma)\) acts faithfully on both biparts. In [9],[19] and [20], a complete classification was given for Subclass (I). In this paper, a partial classification is given for Subclass (II), when \(Aut(\Gamma)\) acts primitively on one bipart.