Researches On Invariants Of Laplacian Matrices Of Graphs

Algebraic Graph Theory is a branch of mathematics that utilizes the algebraic properties of the associated matrices of a graph to study a graph.While the Spectral Graph Theory is an important branch of algebraic graph theory,it investigates the spectra of the associated matrices of a graph and the relations between the spectra and properties of a graph.The study on spectral graph theory has been very active for many years.Many mature and important achievements and applications have also been obtained.A very important part of spectral graph theory is to study the Laplacian matrix of graphs and the related graph invariants.In this thesis,we used graph theory and algebraic methods to study five aspects on graph spectra which include the Laplacian Estrada index of a graph,the graph energy,the Laplacian energy of a graph,the signless Laplacian energy of a graph and the Smith normal form of matrices of a graph.This thesis includes four chapters.The first chapter is an introduction,in which the basic concepts,knowledge of spectral graph theory and the main issues discussed later are briefly introduced.In the second chapter,the connected(n,m)-threshold graphs with max-imal or minimal Laplacian Estrada index are determined,respectively.And the case of disconnected threshold graphs are investigated.Also,it is proved that the connected(n,m)-th.reshold graph with maximal Laplacian Estrada index is the graph Snm proposed in reference[20].At last,it is proved that the connected(n,m)-threshold graph with maximal Laplacian Estrada index is determined by its Laplacian spectrum.In the third diapter,it is proved that for each n ? 3 and p ? 1,(p ? 2 if n = 2),there are n-1 threshold graphs on pn2 vertices,pairwise non-cospectral and equienergetic with the complete graph Kpn2.By this,we unified and generalized the results of[38].Also,we proposed all borderenergetic threshold graphs with n ? 23 vertices and borderenergetic threshold graphs of the form 0p1q0s1t,where p + q + s + t = n,n ? 100.Besides,the Laplacian borderenergetic graphs with minimum or max-imum number of edges are characterised,respectively.And the Laplacian borderenergetic graphs on n ? 10 vertices are determined.Finally,the concept of signless Laplacian borderenergetic graphs is pro-posed.Three families of signless Laplacian borderenergetic graphs are con-structed,the lower bound of the edge number and the extremal number of ver-tices with given number of edges of signless Laplacian borderenergetic graphs are discussed respectively.And the signless Laplacian borderenergetic graphs on n ? 10 vertices are determined.In the fourth chapter,the Smith normal form of the adjacency matrix and distance signless Laplacian matrix of a threshold graph are studied,re-spectively.