On Arc-Transitive Pentavalent Cayley Graphs on Finite Nonabelian Simple Groups

Abstract

A Cayley graph $${\varGamma }=\mathsf{Cay}(G,S)$$ is said to be normal if G is normal in $$\mathsf{Aut}{\varGamma }$$ . The concept of normal Cayley graphs was first proposed by Xu (Discrete Math 182:309–319, 1998) and it plays an important role in determining the full automorphism groups of Cayley graphs. In this paper, we study the normality of connected arc-transitive pentavalent Cayley graphs $${\varGamma }$$ on finite nonabelian simple groups G, where the vertex stabilizer $$\mathsf{A}_v$$ is soluble for $$\mathsf{A}=\mathsf{Aut}{\varGamma }$$ and $$v\in V{\varGamma }$$ . We prove that $${\varGamma }$$ is either normal or $$G=\mathsf{A}_{39}$$ or $$\mathsf{A}_{79}$$ . Further, a connected pentavalent arc-transitive non-normal Cayley graph on $$\mathsf{A}_{79}$$ is constructed. To our knowledge, this is the first known example of pentavalent 3-arc-transitive Cayley graph on finite nonabelian simple group which is non-normal.