Symmetric Graphs Of Order Four Times An Odd Prime Power And Prime Valency

In the field of algebraic graph theory,it is a very important method to use permutation group to described the structure of graph.Let ? be a graph,we denote by Autf(?)its the full automorphism group.If G?Aut(?)acts transitively on arc set,then ? is called G-arc-transitive.An arc-transitive is also called symmetric.if Aut? has no nontrivial normal subgroup N such that ? is a normal cover of the normal quotient graph ?N.The study of symmetric graphs with fixed order is a hot topic.For example,in 1971,Chao had classified symmetric graphs of order p,where p is a prime Then.Cheng and Wang had determined symmetric graphs of order 2p and 3p respectively.Guo etc.had determined the pentavalent symmetric graphs of order 12p,2pn and 2pqn.Note that,Feng etc.had studied the transitive graphs of order 4p;Ghasemi and Zhou had characterized the transitive graphs of order 4p2.There are very few results when n?3.Therefore,it is a very interesting topic to characterize the symmetric graph arc transitive of order 4pn.The main purpose of this thesis is to study symmetric graphs of order four time an odd prime power and prime valency.According to a classical conclusion of the locally-primitive graphs,we discuss in the case of vertex quasiprimitive,vertex bi-quasiprimitive and neither vertex quasiprimitive nor vertex bi-quasiprimitive.and classify all normal quotient graphs of such graphs.Moreover,by using the classification,such graphs with order four times a prime or a prime square are determined.