Researches On The Distance Spectrum And Distance Laplacian Spectrum Of Graphs

Graph theory is a very important scientific, 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 an very important branch of graph theory, over the years of research in the theory of graph spectra has been very active, and has had very mature and im-portant results and applications. In this paper, we combine graph theory with algebraic method and the proposes of matrix theory to study the two specific spectrum:distance Laplace spectra and distance Laplace spectra We obtained some more interesting results on the basis of previous studies with the same time, we solve some conjectures proposed by other researchers. This paper is divided into three chapters, in the first chapter we give the in-troduction, in the second chapter we give some results about the distance spectrum, and in the third chapter we have some results about the distance Laplacian spectrum. In the following, we will give a brief introduction of the three chapter.(i) In Chapter 1, in the first section. We first briefly review the origin of graph theory, and the development process of graph theory, then describes some of the methods and techniques are often used in the research of graph theory. In the second section, we introduce some basic concepts and notations used in this article. There are some special notations, not presented here will make concrete covered in the relevant sections. In the third section, we briefly describe the progress in this paper relates to the issues and problems.(ii) In Chapter 2, in the first section, for a nonnegative irreducible n × n matrix M with diagonal entries 0. we present two sharp upper bounds of the largest eigenvalue of M, also, we give the sufficient and necessary conditions for the equality holds, respectively. As corollaries, we give two sharp upper bounds of the distance matrix of a graph, and characterize the extremal graphs. In the second section, if D is the distance matrix of a graph has n vertices, for a given nonnegative integer k. we show that the (n - k)-th eigenvalue of D λn-k(D)< 1, thereby solving a problem proposed in [41]. In the third section, we characterize the extremal graphs with m-1(D(G))= i, where i= 1,3,4. In the last section, we characterize the sufficient and necessary conditions for complete split graphs to be distance integral,(iii) In Chapter 3, First, we give some lower bounds for the distance Laplacian spectral radius and upper bounds for the second least distance Laplacian eigenvalue of graphs. Second, we show some lower bounds for the distance Laplacian spread of graphs and characterized the extremal graphs. In the last, we prove a sharp upper bound of multiplicity for distance Lapla-cian spectral radius of graphs (not complete graph), and characterize the extremal graphs. thereby confirming the conjecture Aouchiche and Hansen proposed in [3].