On The Spectrum And The Laplacian Spectrum Of A Graph

In a simple graph G = (V,E), let the matrix A(G) = (aij)n×n be the adjacency matrix of G, so the eigenvalues of A(G) are named the eigenvalues of G. The sequence made up of all the eigenvalues is called the spectrum of G. For most graphs, we can't compute their eigenvalues directly, so the estimate of the eigenvalues of graphs is a quite active question for discussion. Over the past 30 years, the articles on the graph problem have been plentiful.Let D(G) be the degree diagonal matrix of G, the matrix L(G) = D(G) - A(G) is called the Laplacian matrix of G, and the eigenvalues of L(G), λ1(G) ≥ λ2(G) ≥…≥ λn(G) ≥ 0, are named the Laplacian eigenvalues of G. The sequence of the n eigenvalues is called the Laplacian spectrum. The study of Laplacian matrix is important for graphs' study because we can estimate many invariants of G, such as connectedness, diameter, bandwidth.The study of the coincidence relation between the spectraldistribution of graph and the structure of a graph can not only hasten the depict of the interrelationship of discrete structure in theory, but also have profound practical background in application, such as the network's optimizes and designs, the design of integrated circuit, and operations research.The main results obtained in this thesis can be summarized as follows:1. Introduce relevant concept and term on the spectrum of a graph and the Laplacian spectrum, and summary the meaning and some advance of the study;2. As tree is an easy and special graph, the study of its eigenvalue can help understand some algebra nature of spectrum. In this article chapter two section four, we will show the upper bounds of trees with perfect matchings.3. The define of the Laplacian matrix of graph has been added the degree of vertices, so the Laplacian eigenvalue can reflect the graph nature even more. In this article chapter three section two, on the basis of the existing conclusions, we will show some upper bounds of the Laplacian spectrum radius using degree of vertices, and show the extreme values, and compare with the existing upper bounds, besides, we show a prove of a given conclusion.