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 $\mathrm{\Gamma }$ with the automorphism group $Aut(\mathrm{\Gamma })$, are divided into two subclasses: (I) $Aut(\mathrm{\Gamma })$ acts unfaithfully on at least one bipart; and (II) $Aut(\mathrm{\Gamma })$ 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 $Aut(\mathrm{\Gamma })$ acts primitively on one bipart.