On Symmetric bi-Cayley Graphs of Prime Valency on Nonabelian Simple Groups

Abstract

Let $\Gamma$ be a bipartite graph, and let $\mathrm{Aut}\Gamma$ be the full automorphism group of the graph $\Gamma$. A subgroup $G\leqslant \mathrm{Aut}\Gamma$ is said to be bi-regular on $\Gamma$ if $G$ preserves the bipartition and acts regularly on both parts of $\Gamma$, while the graph $\Gamma$ is called a bi-Cayley graph of $G$ in this case. A subgroup $X\leqslant \mathrm{Aut} \Gamma$ is said to be bi-quasiprimitive on $\Gamma$ if the bipartition-preserving subgroup of $X$ is a quasiprimitive group on each part of $\Gamma$.
 In this paper, a characterization is given for the connected bi-Cayley graphs on nonabelian simple groups which have prime valency and admit bi-quasiprimitive groups.