The Study Of Eigenvalues Of Some Graphs

The graph spectral theory is a hot topic in the research of graph theory in recent years.It mainly describes the structural properties of graphs though their spectral properties.As important matrixes in the graph spectral theory,the adja-cency matrix,the Laplace matrix and signless Laplace matrix of graphs have a close relationship with the structure of graphs.The main study of graph spectra theory is whether the structural properties of the graph can be reflected by the algebraic properties of the matrix.From the view of the research both from domestic and abroad,the most im-portant research on graph theory focuses on the discussion of some extreme spectral parameters and extreme spectral properties of graphs.The study of the maximal eigeavalues and minimum eigenvalues of various matrices of graphs as well as the problem of characterization the extreme graph have always been the focus of algebra-ic graph theorists.This paper mainly discusses extreme eigenvalues of some graphs with given parameters.It studies the extreme graphs which attain of the maximal spectral radius of graphs with given independence number,and the extreme graphs which attain of the minimum eigenvalues of graphs whose complements graphs with given independence number or given pendent vertices.Specific content arrangements are as follows:In Chapter 1,this paper introduces the background and significance of spectral graph theory,the related symbols and basic conception,and the research problem and research progress of this paper.In Chapter 2,this paper discusses the Laplace spectral radius of the trees with given independence number.In Chapter 3,this paper discusses the least eigenvalue of the graphs whose complements are unicycle graph with n-3 pendent vertexes.In Chapter 4,this paper discusses the least eigenvalue of the graphs whose complements are bicyclic graphs with independence number n-2.