Some Study Of Alternating Forms Graph And Matrices Graph

Algebraic graph theory is an important field of mathematics at home and abroad in recent years, and its two important problems are to characterize the endomorphisms and get a proper coloring of a graph. The geometry of matrices was initiated by famous mathematician Hua L.K.. Recently, some scholars associated them both and studied the properties about matrices graphs, and obtained some results. Based on their work, we will further discuss the properties of the alternating forms graph and the bilinear forms graph in this paper.This paper is divided into three chapters. The first chapter introduces the research background, preliminaries, contents and main results.The second chapter mainly discusses the properties of alternating forms graphs over a finite field IFq. We first verify that both alternating forms graph and bilinear forms graph are not a perfect graph, and discuss the spectra of the lower order alter-nating forms graphs. Then we study an important induced subgraph G’ of alternating forms graph, we obtained that G’ is connected and a Hamilton graph and its chromat-ic number and independence number are qn-2 and gn-1, respectively. We study the independence number and chromatic number of two special graphs, and gain the forms of a maximum independent set.In the last chapter, we discusses the maximal cliques of alternate matrices over a field, and give some conclusions. In section two we obtain some properties about the intersection of two maximal cliques, also we get the algebraic characterization and properties for the intersection of three distinct maximal cliques. These results has sig-nificance for further studying the condition simplification of the fundamental theorem on the matrices geometry and the corresponding graph homomorphism problems.