Some Problems Of Algebraic Graph Theory Related To Geometry

The algebraic graph theory is a branch of mathematics, it is a rapid development direction in recent years. The geometry over a finite field and geometric graph are important geometric structures and combinatorial structures. They involve many fields, such as, association schemes, information science and coding theory, etc. Some scholars study the properties of geometric graphs by connecting some kinds of geometric spaces and graphs,and obtain some results. However, some important problems, for instance, the calculation and estimation of chromatic numbers and independence numbers of some geometric graphs on the finite field ?for example, classical polar graphs, classical dual polar graphs, Grassmann graphs?, haven't been completely solved. In the algebraic graph theory, the research of graph homomorphisms is a core problem. A graph G is called a core if every graph endomorphism of G is graph automorphism. For a graph G, an important question is to judge whether G is a core.This paper is divided into three chapters. The first chapter briefly introduces the research background, preliminary knowledge and main results.In Chapter two, we mainly discuss the properties of the classical polar graphs and classical dual polar graphs, partial geometry and its point graphs. The main results are further distinguishing whether a classical pole graph is a core. The main results obtained in this chapter is theorem 2.1.7, corollary 2.1.9, theorem 2.2.15 and theorem 2.3.4. These results have certain significance to study algebraic graph theory and geometry of matrices.In Chapter three, we mainly discuss the properties of Grassmann graphs. We study the vertex set division of Grassmann graph J_q?4,2?, and the calculation of maximum independent sets of J₃?4,2?.