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

A Cayley graph is said to be normal if G is normal in . 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 on finite nonabelian simple groups G, where the vertex stabilizer is soluble for and . We prove that is either normal or or . Further, a connected pentavalent arc-transitive non-normal Cayley graph on 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.