Some Results On The Signless Laplacian Eigenvalues Of Graphs

Graph theory, a branch of mathematics with widely used, is a powerful tool for discrete mathematics. The theory of graph spectra is an active and important area in graph theory. In graph theory, people introduce various matrices which are naturally associated with a graph, such as the adjacency matrix, the incidence matrix, the distance matrix, the Laplacian matrix, the Signless Laplacian matrix and so on. One of the main problems of graph spectra theory is to determine precisely how, or whether, properties of graphs are reflected in the algebraic properties of the above matrices.Among the above mentioned matrices, the Signless Laplacian matrix can show the properties of graphs more better, since it contains informations about the degrees of all vertices. Cvetkovic, Rowlinson,and Simic [7,8,9,10] discussed the development of the theory of graph spectra based on the Signless Laplacian matrix, and gave several reasons why it is superior to the other graph matrices. This thesis will research the problem of Signless Laplacian matrix. The main content can be divided into three chapters.1. In Chapter1, we first look back the development of graph theory. Then, some definitions and notations for the corresponding questions are given. Lastly, the main results are showed up briefly.2. In Chapter2, we summarize some researchers’ recent studying of the lower bounds on the first, the second and the third Signless Laplacian eigenvalue, and give a lower bound on the Signless Laplacian eigenvalues.3. In Chapter3, we improve the lower bound on the third and the forth Signless Laplacian eigenvalue.