The Genus Of Comaximal Graphs Of Rings And Their Line Graphs

The graph structures on a ring have become increasingly rich in the last three decades since I.Beck introduced zero-divisor graphs of rings in 1988.And these researches have raised a series of interesting questions.The main research object of this paper is the comaximal graph of a ring.Let R be a ring with identity.The comaximal graph of R,denoted by Γ(R),is a graph whose vertices are elements of ring R,and two different vertices a,b are connected if and only if Ra+Rb=R.In this paper,we mainly study the genus of comaximal graphs of rings.If an oriented topological plane Sg is isomorphic to a sphere with g handles,then the genus of the plane Sg is g.The genus of a graph is g if it can be drawn on the directed topological plane Sg but not on Sg-1,such that the intersection of any two edges is only the vertex of the graph.In the first chapter,we introduce the research background,development status and research significance,also,some concepts and basic properties of ring theory and graph theory are introduced.Finally,we give an overview of the main research methods and results of this thesis.In Chapter 2,we study the genus of comaximal graphs.By discussing the number of invertible elements and using the method of graph theory,we determine the classification of finite commutative rings when the genus of comaximal graphs is equal to 0,1 and 2,respectively.Let Γ2(R)be a subgraph of Γ(R)induced by R\U(R)∪J(R)},where U(R)is the unit group of R and J(R)is the Jacobson radical of R.In Chapter 3,we investigate the genus of Γ2(R)based on the restriction of invertible elements of rings on comaximal graphs.In the same way,all finite commutative rings whose genus ofΓ2(R)are 0,1 and 2 are completely characterized,respectively.Based on the results of Chapters 2 and 3,we discuss the line graphs of Γ(R)andΓ2(R)in Chapter 4.All finite commutative rings whose genus of line graphs of Γ(R)and Γ2(R)are 0,1,2 are completely characterized,respectively.