Some Results On The Signless Laplacian Spectrum And Laplacian Spectrum Of Graphs

Graph spectra theory is an important direction in algebraic graph theory. It is mainly related to the eigenvalues and its application, such as the adjacency matrix, incidence matrix, Laplacian matrix and signless Laplacian matrix. All of the eigenvalues of a matrix referred to the graph are called the graph spectrum.In this paper, we mainly discuss the Laplacian spectrum and the signless Laplacian spectrum of graphs. It consists of three parts:the signless Laplacian separator of uni-cyclic and the signless Laplacian separator of bicyclic graphs; the ordering of unicyclic graphs by their signless Laplacian spectral radii and the note on the fourth Laplacian eigenvalues of trees with perfect matchings. The following are the main contents of this thesis.1. In chapter 2, we study the bound of the signless Laplacian separator of uni-cyclic and bicyclic graphs, and determine the graphs which attain the largest signless Laplacian separator.2. In chapter 4, we study the fifth to the seventh largest signless Laplacian sepctral radii among all unicyclic graphs of order n and characterized the corresponding extremal graphs.3. In chapter 5, we study the fourth Laplacian eigenvalues of trees with perfect matchings.