Laplacian Spectral Ratio Of Graphs

The research on graph matrices (such as adjacency matrix, Laplacian matrix, distance matrix etc.) plays an important role in graph theory. The results about the eigenvalues and eigenvectors of graph matrices help people solve various problems in graph theory as well as practical problems formulated as graphs. Among those matrices of graphs, the Laplacian matrix of a graph has been investigated most extensively. Properties on the algebraic connectivity of a graph (defined as the second smallest eigenvalue of the Laplacian matrix of a graph) and the Lapacian spectral radius of a graph have many applications in practical problems. However, we find that in the literature almost all papers focus on only one of the following themes:algebraic connectivity, Laplacian spectral radius, or eigenvectors corresponding to them; only a few of them investigate the ratio of Laplacian eigenvalues.This thesis focuses on the ratio of Laplacian spectral radius and algebraic connectivity of a graph. We call it as the Laplacian ratio of a graph. This ratio reveals in some extension the structure of a graph. It is closely related to the synchronization capability of a network, the smaller the Laplacian ratio, the stronger the synchronization capability. In chapter 2, we determine the Laplacian ratios of complete graphs, stars, paths, cycles, complete multipartite graphs, the union of two complete graphs with some common vertices. Chapter 3 investigates the Laplacian ratios of a graph obtained by attaching some triangles to a single 3-cycles and trees with diameter of 3 and 4. In chapter 4, we propose a conjecture and prove that it is true for some special classes of graphs. The results in this thesis provide a new angle to study the structures and properties of graphs.