On 2-arc Transitive Graphs Of Order Four Times Prime Power

The graphs considered in this thesis are finite connected,undirected simple graphs.A graph ? is called an s-arc transitive graphs,if its full automorphism group Aut? is transitive on the s-arc set of ?.It is easy to know that s-arc transitive regular graphs with s>2 are 2-arc transitive graphs.The study of 2-arc transitive graphs was initialed by Tutte's(1947)famous result:cubic arc transitive graphs are at most 5-arc transitive.Tutte's result was generalized by Weiss[41]by showing that are-transitive graphs with valency greater than 3 are at most 7-arc transitive.Since then,the study of various types of s-arc transitive graphs(2?s?7)has become a very important topic in algebraic graph theory,see[4,5,9,12,14,16,17,19,21,22,24,26,28,31,32,34,36-38].In particular,let p and q be different prime numbers.[42]classified cubic s-arc transitive graphs of order 2p2,[23]characterized 2-arc transitive graphs of order pn,[21]determined affine 2-arc transitive graphs of order pn,[35]classified 2-arc transitive graphs of order pq.The main purpose of this thesis is to characterize 2-arc transitive graphs of order 4pn,where p is an odd prime,and n is a positive integer.In this thesis,we first prove a lemma and introduces some new graphs,and then describe the vertex quasiprimitive and vertex biquasiprimitive case,thus ob-tain the main result(Theorem 1.1),which involves some new examples of graphs.It is worth pointing out that according to Praeger[37],the study of 2-arc tran-sitive graphs can be divides into two steps:the first is to characterize the vertex quasiprimitive and vertex biquasiprimitive case(the obtained graphs are called basic graphs),and the second is to determine the normal covers of the basic graphs.Therefore,the result obtained in this thesis will play an important role for determining general 2-arc transitive graphs of order 4pn.