Cubic And Tetravalent 2-arc-transitive Graphs Of Order Twice An Odd Number

The symmetry of a graph can be described by its automorphism group.A 2arc in a graph is a triple of vertices(u,u,w)such that v is adjacent to both u and w,and u≠w.If the automorphism group of a graph acts transitively on its set of 2-arcs,the graph is called a 2-arc transitive graph.2-arc transitive graphs is a class of highly symmetric graph,have been widely studied in algebraic graph theory.Praeger attributed the study of 2-arc transitive graphs to two steps:studying the basic graphs that are vertex-quasiprimitive or vertex-bi-quasiprimitive,and studying covers of these basic graphs.Researches have extensively studied 2-arc transitive graphs with small degrees and special orders,such as graphs of prime valency with square-free order,cubic and tetravalent with order twice a prime power.In this thesis,we study the 2-arc transitive basic graphs of cubic and tetravalent with order twice an odd integer.By applying the classification theorem for quasiprimitive groups,Aschbacher’s theorem on maximal subgroups of classical groups,and coset graph theory,we completely classify the basic graphs that are vertex-quasiprimitive or vertex-biquasiprimitive,and construct six infinite families of new 2-arc transitive graphs,of which three are vertex-quasiprimitive on three-dimension linear groups and unitary groups,and the other three are vertex-bi-quasiprimitive on two-dimension linear groups.In particular,the 2-arc transitive cubic graph on three-dimension linear group that we construct is a previously overlooked class of graphs.