On biprimitive semisymmetric graphs

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