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

Graph theory and algebra are two important branches of mathematics.Inclusion graph,as one of the intersections of the two,is of great significance in the fields of group theory,ring theory and so on.We assume that Fq is a finite field with q elements,where q is a power of an odd prime.And O is called the 2?+? dimensional orthogonal space over Fq with respect to S2?+?,?,where ?=0,1 or 2.In this paper,we introduce a family of totally isotropic subspace inclusion graphs on an orthogonal space O of odd characteristic,denoted by In(O),where the vertex set is the collection of totally isotropic subspaces with dimension at least 1 and two vertices O1 and O2 are adjacent if either O1?O2 or O2?O1.Compared with the subspace inclusion graph of a vector space In(V)studied by Das,the graph In(O)introduced in this paper is the induced subgraph of In(V).Due to the limitation of vertex set,the results and the research methods in this paper will be different from the former.This thesis mainly studies the structure and properties of the graph In(V).In chapter 1,we introduce some preliminaries on the orthogonal space of odd characteristic and the graph.In chapter 2,we give the calculation formulas of the number of vertices and the degree of vertices of In(O).In chapter 3,we discuss the diameter,girth,clique number,chromatic number,triangulation of In(O)and the necessary and sufficient conditions for it to be Eulerian,and prove that the isomorphism of two orthogonal spaces is equivalent to the isomorphism of their corresponding totally isotropic subspace inclusion graphs.In addition,for the graph In(O)with special parameters,we study its planarity.In chapter 4,we prove the necessary and sufficient conditions for In(O)to be bipartite,regular graph and Deza graph respectively,and give a concrete example of In(O).