Arc-transitive Cubic Graphs Of Order4pq

An important method for studying structures of graphs in algebraic graph theo-ry is using group theory, especially using permutation group theory. Characterizing symmetry of graphs is an important topic in algebraic graph theory, which is main-ly depending on the characterization of actions of the full automorphism groups on graphs. Let Γ be a graph, and let AutΓ denote its full automorphism group. If AutΓ is transitive on the arc-set of AΓ, then Γ is called an arc-transitive graph. The study of finite arc graphs was first initialed by Tutte, and then many nice results were obtained. For example, Conder etc. determines all cubic arc-transitive graphs with order up to768in2002; Xu etc. classified cubic arc-transitive graphs of order4p with p a prime in2004; by using graph covering theory, Feng and Kwak classified cubic arc-transitive graphs of order4p2in2007; and Feng etc. classified cubic arc transitive graphs of order np and np2with4≤n≤10in2007. For more results, refer to [7,11].The main aim of this thesis is to characterize arc transitive cubic graphs of order4pq, where p>q≥5are prime. The main method used is to analyze the algebraic structures of the corresponding arc-transitive automorphism groups and their actions on the graphs. We divide the discussion into two cases depending on that if the corresponding automorphism groups contain a soluble normal subgroups or not. For the former case, by analyzing normal quotient graphs and using covering theory, the graphs are characterized; For the latter case,we prove that the automorphism groups are almost simple and specifically determined, by analyzing their orbital graphs, the graphs are classified.