Totally Isotropic Subspace Sum Graph Of An Orthogonal Space Of Odd Characteristic

Algebraic graph theory is an important branch of discrete mathematics.It mainly uses algebraic method to study some properties of graph,which enriches the knowledge of graph theory.In this paper,by combining graph theory and orthogonal space,we define the totally isotropic subspace sum graph of an orthogonal space of odd characteristic,and study some properties of the graph by using the architectural features of the orthogonal space.Let F_q be a finite field of q elements with odd characteristic,and O be a 2v+δ（δ=0,1,2）-dimensional orthogonal space over F_q.In this paper,we define the totally isotropic subspace sum graph of an orthogonal space of odd characteristic,denoted by g（O）,its vertex set is the set composed of all non-zero and non-maximum totally isotropic subspaces in O;For any two vertices O₁and O₂in g（O）,if the sum of O₁and O₂is a maximal totally isotropic subspaces,then the vertices O₁and O₂are said to be adjacent.In（V）,the subspace sum graph of a vector space V,any two vertices are adjacent if and only if the sum of these two vertices is the vector space V.But in g（O）,any two vertices are adjacent if and only if the sum of these two vertices is maximal totally isotropic subspace.However,maximal totally isotropic subspace is not unique in orthogonal space.Therefore,the method used in the study of graph g（O）is different from that used in g（V）.In chapter 1,we introduce some definitions and theorems of orthogonal spaces and graph theory.In chapter 2,we give the definition of the totally isotropic subspace sum graph of an orthogonal space of odd characteristic,and the totally isotropic subspace sum graph of 4-dimensional orthogonal space over F₃are studied.Next,we study some properties of graph g（O）,including diameter,girth,clique number and a maximal inde-pendent set.Then we obtain the sufficient and necessary conditions for isomorphism of two totally isotropic subspace sum graphs.In chapter 4,we give the calculation formula of degree of vertex,and according to this formula,it is proved that the graph g（O）is a regular graph only when v=2.In addition,it is proved that the graph g（O）is an Euler graph,not a planar graphs or a complete graph.