Semisymmetric graphs of order $2p^3$

TL;DR

This paper investigates semisymmetric graphs of order 2p^3, aiming to classify those where the automorphism group acts unfaithfully on at least one bipart, advancing understanding of these complex symmetric structures.

Contribution

It provides a group theoretical characterization for a subclass of semisymmetric graphs of order 2p^3, laying groundwork for a complete classification of these graphs.

Findings

Characterization of automorphism group actions on biparts

Division of graphs into two subclasses based on group action

Foundation for future complete classification of semisymmetric graphs

Abstract

A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin. Theory Ser. B} {\bf 3}(1967), 215-232] that there exist no semisymmetric graphs of order $2p$ and $2p^2$, where $p$ is a prime. The classification of semisymmetric graphs of order $2pq$ was given in [{\em Comm. in Algebra} {\bf 28}(2000), 2685-2715], for any distinct primes $p$ and $q$. Our long term goal is to determine all the semisymmetric graphs of order $2p^3$, for any prime $p$. All these graphs $\G$ are divided into two subclasses: (I) $\Aut(\G)$ acts unfaithfully on at least one bipart; and (II) $\Aut(\G)$ acts faithfully on both biparts. This paper gives a group theoretical characterization for Subclass (I) and based on this characterization, we shall give a complete classification for this subclass in our further research.