On The Distance Laplacian And Distance Signless Laplacian Spectra Of Graphs

The concept of the distance Laplace matrix GL)( and the distance signless Laplace matrix GQ)( of a graph is proposed by M.Aouchiche and P.Hansen in the paper “Two Laplaicans for the Distance Matrix of a Graph” in 2013. In this paper,the authors defined the L -=GDGTr G)()()( and Q +=GDGTr G)()()(,here the GD)( the distance matrix of G and GTr)( the diagonal matrix of the vertex transmissions in G.The distance Laplacian and the distance signless Laplacian eigenvalues of graphs are called the distance Laplacian and the distance signless Laplacian spectra.In this paper, we mainly study the distance signless Laplacian eigenvalues and the distance Laplacian and the distance signless laplacian spectral radius.In chapter 1, we review the development of Graph Theory firstly. Then we introduce the backgrounds and statues of this paper and some basic definitions of Graph Theory.In chapter 2, we study the distance signless Laplacian eigenvalues of graphs.Concretely we discuss the inequalities on the distance signless Laplacian eigenvalues of graphs and the properties of the distance signless Laplacian eigenvalues of graphs.In chapter 3, we determine the trees with given diameter,the unicyclic and bicyclic graphs with given girth and the connected graphs with cut-edge having the minimum distance signless Laplacian spectral radius. Moreover we introduce the influence on the distance signless spectral radius of graphs when subdividing or contracting an edge.In chapter 4, we determine the trees with n-3 pendent vertices, the connected graphs with cut-vertex or cut-edge having the minimum distance Laplacian spectral radius.In chapter 5, we primarily summarize the main conclusions of this paper and give some considerable problems for further study.