On Laplace Eigenvalues Of Graph

Let G be a simple graph and A(G)be the adjance matrix of G, eigenvalues and spectrum of A(G)are called eigenvalues and spectrum of G. The corresponding relation of the spectrum graphs and the structures of graphs is an active research direction. Acorrdingly, people introduce the Laplacian spectrum of G, the Laplacian matrix of G is L (G) = D(G)-A (G),where D(g) is the diagonal matrix of vertex degrees. The Laplacian spectrum is denoted by S (G) = (λ1,λ2, ..., λn). There are abundance of related literature and results, which reflectplenty of structures of graphs. So people more and more pay attention to Laplacian spectrum of graphs. This research not only intensifies the description of inherence relation of discrete structures, but also has actually far-reaching meaning in network抯optimizing and design, IC抯 design, operational research, and so on.In the research of Laplace spectrum, the largest Laplace eigenvalue is very important. We mainly estimate its upper bound and determine structures of graphs when it is equal to its upper bound. Thinking of this way, people try to research upper bounds of other eigenvalues. Howerer, these results are not easily received.Besides, Laplace eigenvalues can estimate a good many invariables of graphs, such as connectivity, diameter, bandwidth, so Laplace matrix is very important to the research of graph theory.This paper gives some new and more accurate results on upper bounds of the largest Laplace eigenvalues and its main contents are the following:In the first chapter, we introduce the basic theory of graphs;In the second chapter, we review the Laplace eigenvalues of graphs, especially the development of the largest Laplace eigenvalues and introduce some upper bounds of the largest Laplacian eigenvalues of graphs;In the third chapter, we introduce the second largest Laplace eigenvalues and degree of graphs;In the fourth paper, we introduce the related invariables of graphs of Laplace eigenvalues of graphs.