The Quotientability Of Arc-Ransitive Graph And Normal Edge-Transitive Cayley Graphs

The symmetry of graphs be a key subject in the study of groups and graphs all the time. We called a graph is symmetric if its automorphism group acts on the arc set of the graph transitively. The automorphism of special orbit graph is connected with the automorphism graph of original graph, and so we can study the original graph through the quotient graph. Because of the particularity of vertex set and edge set of Cayley graph, the structure of automorphism group is special too. It is well known that the automorphism group is clear when the Cayley graph is normal. For the graph Cay(G,S) is normal if and only if G is normal in Aut(Γ), then Aut(Γ)= R(G)Aut(G,S). But it is important and difficult to judge the Cayley graph is normal or not. And so we defined the normality of edge-transitive, namely, N Aut(T)(G) transitively acts on the edge set E(Γ), for N Aut(Γ)(G)＝R(G)Aut(G,S).So we study the quotientability of symmetric graph in the third chapter of the paper. For a arc-transitive graph F, if it is regular covering graph of a simple arc-transitive graph, we called it is quotientable, otherwise, it is basic. In the chapter, we obtain the classification the quotientability of arc-transitive graphs with va- lency prime in the paper. Furthermore, according to the classify of pentavalent arc-transitive which has been given, we classify the quotientability of pentavalent arc-transitive graph in another way.Let Γ=Cay(G,S) be a Cayley graph, if N Aut(Γ)(G) acts on edge set is transi-tive, we called Γ is normal edge-transitive. In the fourth chapter of the paper, we get a classification of the normal edge-transitive with order pq (p, q prime, and p> q> 2). Up to isomorphism, there is only one normal edge-transitive Cayley graph of valency 2p. There are at most q-1 normal edge-transitive Cayley graphs of valency 2d for d|p-1. In particular, there are q-1 normal edge-transitive Cayley graphs when d≤q-1/2.