The Inverses Of Distance Matrices Of Some Graphs

Research of matrices of a graph is an important topic in algebraic graph theory, and there are many related results on matrices and graph.In the 1970 s, R. L. Graham and H. O. Pollak [17] gave the determinant and inverse of matrices for distance matrix of a tree. In recent years, R. B.Bapat et.al. generalized the relevant results of trees, got the expressions of determinants and inverses of distance matrices for some special graphs.For a tree, all of the blocks are K2(a graph which block is complete graph),so it is a block graph. This thesis mainly studied the distance matrix of graphs which blocks are complete bipartite graphs or cliques. The results of this thesis are generalization of related results for trees.This thesis is divided into four chapters.In chapter 1, we introduce the research background and some necessary preparation knowledge which will be used.In chapter 2, we give the determinant as well as the inverse expression of the distance matrix in bi-block graph. Let G be a bi-block graph on N vertices with r blocks Kmi,ni, 1 ≤ i ≤ r,thenIn chapter 3, we discuss the determinant,inverse matrix and Smith normal form of the q-distance matrices of block graph. Let G be a block graph on n vertices with r blocks Kn1, Kn2, · · ·, Knr,where q ≠±1, thenIn chapter 4, we summarize the main results of thesis and propose the future research problems.