The Study On The Eigenvalues Of Graphs

Properties of graphs are closely associated with the spectra of some matrices, such as the adjacency matrix, the distance matrix, the(signless) Laplacian matrix, etc. One important topic of the theory of graph spectra is how to determine the properties of graphs by the algebraic properties about eigenvalues(spectral radius, spread, energy, etc) of such matrices.Among all the matrices of graphs mentioned above, the adjacency and Laplacian matrices of graphs have been much more investigated in the past. Recently, the spectra of the signless Laplacian matrix of graphs becomes an active and important area in the theory of graph spectra.Cvetkovi′c gave some suggestions for twelve further investigations of the theory of graph spectra, one of them was classification and ordering of graphs. Based on this idea,people have done much investigation on the algebraic properties of different matrices of graphs, which was fixed some variable.In this paper, we mainly investigate the maximum eigenvalues(spectral radius) of adjacency or signless Laplacian matrices of graphs with given some variable, and try to order them. The contents of this thesis as follows:In Chapter 1, we first look back the evolvement of the graph theory. Then, we introduce the backgrounds and research progress about the theory of graph spectra. Finally,we introduce some notations and definitions of the theory of graph spectra.In Chapter 2, we study the spectral radius for adjacency matrices of graphs with given clique number. We obtain the first four smallest values of the spectral radius among all connected graphs with clique number ω ≥ 2, and give the corresponding graphs.In Chapter 3, we study the signless Laplacian spectral radius of graphs with given clique number. And we obtain the first four smallest values of the signless Laplacian spectral radius among all connected graphs with clique number ω ≥ 2, and give the corresponding graphs.In Chapter 4, we study the signless Laplacian spectral radius of unicyclic graphs with given matching number, and determine the first four(three) largest signless Laplacian spectral radius of unicyclic graphs with fixed matching number μ ≥ 3(= 2).In Chapter 5, we study the signless Laplacian spectral radius of bicyclic graphs with perfect matchings, and determine the largest signless Laplacian spectral radius of bicyclic graphs with perfect matchings.In Chapter 6, we determine the unique graph with the largest signless Laplacian spectral radius among all the tricyclic graphs with n vertices and k pendant vertices.