Study Of Some Problems For Algebraic Graphs And Geometry Of Matrices

Algebraic graph theory is a booming direction of mathematics. It studies the problems of graph theory by using algebraic method. Geometry of matrices is a field of mathematics which was initiated by academician Hua L.-K. in forty years of twentieth century. In recent years, the study of algebraic graph theory and matrix geometric are intersecting and permeating. In this paper, we respectively discuss the fundamental theorem of the geometry of centre-symmetry matrices (centre-alternating matrices), the problem of chromatic number and independence number for symmetric bilinear forms graphs and Hermitian forms graphs.This paper is divided into four chapters. In Chapter one, we introduc the back-ground of this topic, research situation and main results.In Chapter two, we prove the fundamental theorem of the geometry of centre-symmetry matrices (centre-alternating matrices) by applying the fundamental theorem of the geometry of symmetric matrices (alternate matrices). Although the proof is relatively simple, but they are new results and useful supplement to the study of matric geometric.In Chapter three, we discuss the property of symmetric bilinear forms graph. First, we calculate the spectra of symmetric bilinear form of graph when n = 2,3 and q is even. Then, we prove that the independence number and the chromatic number of the symmetric bilinear forms graph Sym(3,2) is 22 and 4, respectively.In the first section of Chapter four, we show that the chromatic number of Hermi-tian forms graph is> q+1. In second section of Chapter four, we mainly discuss the independence number of Hermitian forms graph Her(2,3~2). We found that the upper bounds on the independence number are too large by using known inequality. By us-ing orthogonal similarity class, we obtain that the independence number of Hermitian forms graph Her(2,3~2) is 15, which is helpful to the further study of the independence number of Hermitian forms graphs.