Research On Sereval Problems Of The Algebraic Properties Of Simple Graph

Graph theory takes graph as the research object,and represents the concrete or things in life with points,while some specific relationship between things is represented with the connection between points.In recent decades,graph theory has been widely developed and applied,and graph theory has been well applied in chemistry,physics,integrated circuits and other disciplines.The relationship between points and lines in graph theory can be described by matrix.The relationship between eigenvalues and eigenvectors of matrix can reflect some characteristics of graph.In 1957,Collatz got the conclusion that the zero dimension of a tree is closely related to its structural properties,and that the rank of a tree must be even.In this paper,we study the zero dimension set of single cycle graph[47],double cycle graph[48,49],and three cycle graph[50],as well as the problem of zero dimension of graphs with special structure.Chang An et al.Investigated the properties of nonsingular graphs and solved the related problems of simple graphs and nonsingular graphs with the idea of algebraic graph theory.In this paper,based on the previous research results,we characterize graphs with rank 7 and no three and four cycles.We study the concrete graphs of nonsingular graphs and simple graphs with rank 7.Let F be an algebraically closed field of characteristic zero,and letsp(2l,F)be the rank 1 symplectic algebra over F of all(2l)×(2l))matrices(?)to over F,where At is the transpose of A,and B,C are both symmetric matrices of order l.The commuting graph Γ(sp(2l,F))of sp(2l,F)is a graph whose vertex set consists of all nonzero elements in sp(21,F),and two distinct vertices x and y are adjacent if and only if xy=yx,or equivelently,the bracket product[x,y]＝xy-yx of x and y is zero.In this article,the diameter of F(sp(2l,F))is proved to be 4 when l>2.The paper is divided into six chapters.In Chapter 1,introduces the background of the topic and the current situation of development.In Chapter 2,introduces the basic concepts used in this paper and its proof and other basic knowledge.In Chapter 3,we introduce the study of the rank of simple graphs.In Chapter 4,we characterize a reduced graph of rank 7 without three or four cycles.In Chapter 5,introduce the diameter of commutative graphs on symplectic algebras.In Chapter 6,summary and problems to be solved.