Laplacian Spectrum And Laplacian Coefficients Of Graphs

The spectral graph theory is one of the main topics in algebraic graph theory,which chiefly concerned with the adjacency spectrum,the Laplacian spectrum and the signless Laplacian spec-trum of a graph.By the matrix representation of a graph,the spectral graph theory mainly studies the relationship between the spectrum and structures of graphs,discuss how the structural proper-ty of a graph is characterized by its spectral property.The theory of graph polynomials is another main branch in algebraic graph theory.Since graph invariants are contained in coefficients of graph polynomials,people usually use theory of polynomials to study properties of graphs.Both the spectral graph theory and the theory of graph polynomials have some important application-s.In this thesis,we mainly considers two kinds of problems.One is the extremal problem on Laplacian spectral radius and Laplacian Estrada index in some given sets of graphs,and the other is the asymptotic normality of Laplacian coefficients.Specific contents are as follows:The first part considers the extremal problem on Laplacian spectral radius of all trees with n vertices and diameter 4.By the structural properties of these trees and the eigenvector equations,we characterize the extremal graph with minimum Laplacian spectral radius.The second part primarily studies the extremal problem on Laplacian spectral radius of all unicyclic graphs with n vertices and fixed diameter d.By edge grafting transforms and classic results on the bounds of the Laplacian spectral radius,we determine the corresponding graph with maximal Laplacian spectral radius.The third part researches the extremal problem on the Laplacian Estrada index of connected graphs.For connected graphs with n vertices and m edges,where 3n-5/2<m?2n-6,n?7,and connected graphs with n vertices and r pendant vertices,we obtain the upper bound and the corresponding extremal graph,respectively.The fourth part devotes to the asymptotic normality of Laplacian coefficients of graphs.we give some methods to determine the asymptotic normality of Laplacian coefficients of some graphs.Such graphs include the paths,the binary trees,the stars,the cycles,the wheels,regular graphs as well as the hypercubes.