Research On Local Structure And Symmetry Of 2-arc Regular Graph

The classification or characterization of graphs with various transitive properties is an important and active scientific research topic in algebraic graph theory.In this paper,we mainly study 2-arc regular graphs,especially on 2-arc regular graphs of square-free order.A graph is called a 2-arc regular graph if its fully automorphism group is regular on its 2-arc set.If a positive integer is not divisible by the square of any prime number,it is called a square-free number.Based on the classification results of finite simple groups and the structure of almost simple groups,this paper explores and classifies the properties of Sylow subgroups,and obtains a classification result of 2-arc regular graphs of square-free order that allow almost simple groups.When investigating the local structure of the above graph,it is found that whether the stable subgroups exchange has a certain influence on the automorphism group structure of the graph,so this paper also explores the generalized exchange properties of the elements in the group(ring),and obtains some special properties characterization of the ring and its extensions nature.This paper is divided into three chapters.The introduction of the first chapter briefly introduces the preliminary knowledge,research background,significance and main work used in this article.The second chapter mainly study the classification of(G,2)-arc regular graphs of square-free order,where G is an almost simple group.According to the classification results of finite simple groups,the structure of almost simple groups G is restricted to specific groups by two theorems,and the structure of point stable subgroup Ga is obtained,and then the existence of special elements and the corresponding 2-arc regular graphs ?=Cos(G,G?,G?gG?)are constructed.The classification results of 2-arc regular graphs of square-free order are obtained.The third chapter mainly study the weakly reflexive rings with an involution.We further study the weakly reflexive rings by introducing the concept of weakly reflexive rings with an involution.The equivalent conditions and several properties of weakly reflexive rings with an involution are given,and four typical ring extensions,such as trivial extension,Dorroh extension are discussed.