| Graph theory is an active and important branch in applied mathematics,it is widely used in various fields,such as computer networks,life sciences,biochemistry,combinatorial optimization,molecular theory.And the theory of graph spectra is a research hot spot in graph theory.In 1971,Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems.They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only.This impressive result made distance matrix spectral properties a research subject of great interest.In this paper,we combine graph theory with algebraic method and the pro-poses of matrix theory to study the distance spectra of graphs and weighted trees.We obtain some interesting results on the basis of previous studies.This paper is divided into six chapters.In the first chapter,we give the in-troduction.In the second chapter,we consider the connected graphs with?n?D?G???[-(1+?17?1/2)/2),-1-21/2).In the third and fourth chapters,we con-sider the multiplicity of eigenvalues-2 and 0 of the distance matrix of graphs,respectively.In the fifth chapter,we mainly consider the distance between dis-tance spectra of special classes of graphs.In the sixth chapter,we give some results about distance spectral radius of weighted trees.In the following,we will give a brief introduction of the six chapters.In chapter 1,we first look back the evolvement of graph theory.In partic-ular,we give a brief introduction of the distance matrix of graphs.Then,we introduce some basic concepts and notations used in this article.There are some special notations,not presented here will give a detailed introduction in the relevant chapters.In chapter 2,we first introduce the research background.In the second sec-tion,we provide some useful results that will be used throughout the chapter.In the third section,we characterize the graphs with?n?D?G???[-(1+?17?1/2)/2,?-1)U[?-1,-1-21/2),where ? is the smallest root of x3-x2-3x + 1 = 0,-(1+?17?1/2)/2??-1?-1-21/2and Furthermore,we show that the graphs with ?n?-(1+?17?1/2)/2 are determined by their D-spectrum.In chapter 3,we characterize the graphs with m-2?D?G??=n-i,where i = 1,2,3,4.Furthermore,we show that bothSn+ and Sa,b?a+b=n-2? are determined by their D-spectrum.In chapter 4,we first introduce the research background and motivation.Then,we characterize graphs with n0?D?G??= n-i,where i=1,2,3,4.Furthermore,the graphs are determined by their D-spectrum.In chapter 5,we first introduce the research background.Secondly,we obtain lower bounds on ??G,Kn?and ?(G,Ka,b)for a + b = n.Furthermore,we give an upper bound on csn.In chapter 6,we first introduce the research background.In the second section,we determine the weighted trees with the smallest distance spectral radius and the second smallest distalce spectral radius in TW,where W ={W1,W2…Wn-1},where W1?W2?…?Wn-1?0. |
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