On biprimitive semisymmetric graphs

Abstract

A regular bipartite graph $\hat{I}“$ is called semisymmetric if its full automorphism group $\mathrm{Aut}(\hat{I}“)$ acts transitively on the edge set but not on the vertex set. For a subgroup $G$ of $\mathrm{Aut}(\hat{I}“)$ that stabilizes the biparts of $\hat{I}“$, we say that $\hat{I}“$ 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.

Your kind suggestions and insights are most warmly welcomed!