Symmetric bi-Cayley graphs on nonabelian simple groups with prime valency

Abstract

Yin et al. (2021) [19] classified connected arc-transitive Cayley graphs on nonabelian simple groups with prime valency p≥11 and solvable vertex stabilizers. In this paper, we extend this result to bi-Cayley graphs. Let Γ be a connected arc-transitive bi-Cayley graph on a nonabelian simple group T with prime valency p≥5 and solvable vertex stabilizer. It is proved that either Γ is nonbipartite, or Γ is a normal bi-Cayley graph, or Γ is S-semisymmetric with (S,T,p)=(PSL2(11),A5,11),(PSL2(29),A5,29),(M23,M22,23), or (An,An−1,p) where p|n|pkl and k|l|(p−1). Moreover, example exists for each triple (S,T,p) above.