The Contiguous Spectrum Of The Graph And The Study Of The Unsigned Laplacian Spectrum

The graph spectra theory is an active and important research area in graph theory. One of the main problems in the theory of graph spectra is to determine precisely how and whether, properties of graphs are reflected in the algebraic properties of the above matrices.This thesis mainly investigates the spectral radii for adjacency matrices and signless Laplacian matrices of a graph. We get main conclusions by using characteristic polynomial methods and Zero-point Theorem as follows.In the third chapter, we focus on adjacency spectral radius of the graph. In the first section we give a few representative conclusions about adjacency spectrum in recent years, including using the number of edges or diameter or coloring number and so on to characterized the spectral radius. In the second section, we use some bounds of spectral radius and obtain the first 9 largest spectral radius of graphs, that is, ?(Kn)>?(Kn-K2)>?(Kn-P3)>?(Kn-2K2)>?(Kn-K1,3)>?(Kn-C3)> ?(Kn-P4)>?(Kn-P3?K2)>?(Kn-3K2).In the fourth chapter, we address the weighted graph, and mainly focus on the bounds of its signless Laplacian spectral radius. We get the upper bounds of the weighted graphs and find the extreme graph.