On biprimitive semisymmetric graphs

TL;DR

This paper introduces a method to construct and classify biprimitive semisymmetric graphs with almost simple automorphism groups, focusing on the symmetric and alternating groups, and provides a detailed subgroup analysis.

Contribution

It offers a new construction technique for biprimitive semisymmetric graphs and classifies such graphs when the automorphism group is A_n or S_n.

Findings

Constructed infinite families of biprimitive semisymmetric graphs

Classified graphs with automorphism groups A_n or S_n

Identified pairs of maximal subgroups with same order

Abstract

A regular bipartite graph $\Gamma$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\Gamma)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\Gamma)$ that stabilizes the biparts of $\Gamma$, we say that $\Gamma$ is $G$-biprimitive if $G$ acts primitively on each part. In this paper, we first provide a method to construct infinite families of biprimitive semisymmetric graphs admitting almost simple groups. With the aid of this result, a classification of $G$-biprimitive semisymmetric graphs is obtained for $G=\mathrm{A}_n$ or $\mathrm{S}_n$. In pursuit of this goal, we determine all pairs of maximal subgroups of $\mathrm{A}_n$ or $\mathrm{S}_n$ with the same order and all pairs of almost simple groups of the same order.