Study Of Some Problems For Matrices Graphs And Matrices Geometry

The study of geometry of matrices was initiated by Hua L.-K. in forties of the last century. A geometry of matrices corresponds to a connected graph G=(V,—), where-is an adjacency relation. In 2009, M. Orel showed that any Hermitian forms graph is a core. In 2012, M. Orel proved that any symmetric bilinear forms graph Sym(n, q) is a core whenever n> 2, and computed the eigenvalues of Sym(n,q). Based on these work, we will further study some properties for Hermitian forms graph and symmetric bilinear forms graph.This paper has four chapters. In Chapter 1, we give a brief introduction of the background, present research situation and main results. In Chapter 2, we discuss the perfection of symmetric bilinear forms graph. The main results are:the symmetric bilinear forms graph Sym(2,2) is a perfect graph, but when Sym(n, q)≠Sym(2,2) Sym(n,q) is not perfect. Then we discuss the independence number and chromatic number for Hermitian forms graph and symmetric bilinear forms graph, and show that the chromatic number of the Hermitian forms graph Her(2,22) is 4, which has reference value for further discussing the chromatic number of a Hermitian forms graph. We also calculate the spectra of the lower order Hermitian forms graphs.In Chapter 4, we discuss the graph endomorphisms about the symmetric bilinear forms graph Sym(2,q). We explain that both Sym(2,2) and Sym(2,3) are not a pseudo-core. In order to study whether Sym(2,q)(q> 4) is a pseudo-core, we prove the following results. Let q=pm be an odd number≥5 where p is a prime. If φ is a graph endomorphism on Sym(2, q) such that φmaps two distinct maximum cliques which non empty intersection onto two distinct maximum cliques, then φ is an automorphism.