Research On Some Related Problems Of The Edge-primitive Graphs

It is an important method of algebraic graph theory to use group theory,especially permutation group theory to study graph construction.The symmetry of graphs is an important research topic in algebraic graph theory.The symmetry of a graph is mainly described by the function of the automorphism group of the graph on various objects of the graph,such as the transitive or primitive properties of the full automorphism group on the vertice set,edge set and -arc set of the graph.Let Γ be a finite simple undirected graph whose full automorphism group is called Aut（Γ）.If Aut（Γ）is transitive on the edge set of Γ,then Γ is said to be an edge-transitive graph.The study of edge-transitive graphs is a hot topic in algebraic graph theory.If the action of Aut（Γ）on the edge set of Aut（Γ）is not only transitive but also primitive,then Γ is said to be edge primitive graph.Edge primitive graph is an important class of arc-transitive graphs,which was studied by the famous algebraic graph theorist Weiss in 1973.This type of graph is subject to extremely strict conditions,but some famous graphs such as Heawood graph,Tutte-Coxeter graph and Higman-Sims graph are edge-primitive graphs.In this paper,we mainly study the related problems of edge-primitive graph.A general study of edge-primitive graph began in 2003.In the past ten years,the research of edge-primitive graph has achieved some important results.In this paper,we study several classes of edge-primitive graphs by constructing coset graphs.We first study the classification results of vertex-primitive 2-path-transitive graphs and bivertex-primitive2-path-transitive graphs,and complete the general characterization of edge-primitive 2-path-transitive graphs.Secondly,we give the classification of edge-primitive graphs on almost simple groups whose socle are Mathieu groups.Surprisingly,there are quite a few edge-primitive graphs on Mathieu groups.Finally,we determine the complete classification of edge-primitive graphs of order by using the completed classification of simple groups with an index subgroup,where and are odd prime numbers.