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

Let Γ be a bipartite graph, and let AutΓ be the full automorphism group of the graph Γ. A subgroup G AutΓ is said to be bi-regular on Γ if G preserves the bipartition and acts regularly on both parts of Γ, while the graph Γ is called a bi-Cayley graph of G in this case. A subgroup X Aut Γ is said to be bi-quasiprimitive on Γ if the bipartition-preserving subgroup of X is a quasiprimitive group on each part of Γ. 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.