Some Research On The Eigenvalues Of Graphs

The theory of graph spectra is an active and important area in graph the-ory. There are extensive applications in the fields of quantum chemistry, statistical mechanics, computer science, communication networks and information sci-ence. In the theory of graph spectra, there are various matrices that are nat-urally associated with a graph, such as the adjacency matrix, the (signless) Laplacian matrix, the incidence 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 of graphs, the most important two are the adjacency matrices and the Laplacian matrices of graphs. This thesis mainly investigates the limit points of the Laplacian spectra of graphs, spectral radius of digraphs with given diameter and sharp bounds on the signless Laplacian spectral radius of weak joining graphs. The main content of this thesis are as follows.(ⅰ) In Chapter 1, we first look back the evolvement of graph theory. Then, we introduce some definitions and notations for the corresponding questions. Lastly, we introduce the backgrounds and research progresses of some questions on graph spectra theory which we study in this paper.(ⅱ) In Chapter 2, firstly,we present a sequence of graphs{Gn} with by investigating the eigenvalues of the line graphs of {Gn}. Moreover, we prove that 1 and this limit are the minimal and second minimal limit points of the third largest Laplacian eigenvalues of graphs. In the second section, let l3(b) and l'3(b) be the second largest root of 6μ(μ-2)-(μ-1)2(μ-3)= 0 and bμ(μ-2)-(μ-1)2(μ-3)-(μ-1)(μ-2)= 0, respectively. Firstly, we will prove that both l3(b) and l'3(b) are the limit points of the third largest Laplacian eigenvalues of graphs and obtain that l'3(b)