Arc-transitive Normal Cayley Graphs Of Order4p~2and Prime Valency

In this thesis, graphs are assumed to be connected, undirected and simple. A graph is called arc-transitive if its full automorphism group is transitive on its arc set. A graph Γ is called a Cayley graph of a group G, if there is a subset S of S (?) G\{1} such that S=S-1:-{g-1|g∈S}, the vertex set of Γ is G, and two vertices x and y are adjacent if and only if yx-1∈S. Suppose Γ=Cay(G,S) is X-arc-transitive, if G(?)X≤AutΓ,then r is called an X-normal arc-transitive Cayley graph. In particular, if G(?) AutΓ,then r is called a normal Cayley graph.Normal Cayley graphs form an important class of transitive graphs. Since they are closely related to their full automorphism groups, and characterizing the full automorphism groups of graphs is a fundamental topic in algebraic graph theory and is generally very difficult, normal Cayley graphs have received much attention in the literature. For example, Yu hui xia proved that tetravalent1-arc-regular Cayley graphs of order2p2with p an odd prime are normal in two thousand five; Feng yan quan shown that tetravalent arc-transitive Cayley graphs of order p3with p an odd prime are normal in two thousand five; Zhou yong an characterized tetravalent Cayley graphs of generalized quaternion groups in two thousand senve; and Darafsen and Assari characterized edge-transitive noemal Cayley graphs of order four times a prime in two thousand thirteen.By analyzing the structures of groups of order4p2with p a prime, characterizing the automorphism groups of the group, investigating the properties of normal arc-transitive Cayley graphs of order4p2, it is proved that there is no normal arc-transitive prime valent Cayley graphs of order4p2.