On The Normalized Laplacian Spectra And Characteristic Polynomials Of Graphs

In recent years,the theory of graph spectra has been widely used in physics,chemistry and computer network,ect.The main research problem of spectral graph theory is the relationship between the eigenvalues and structures of graphs.These eigenvalues have a close relationship with other graph invariants,and it is necessary to obtain the structural information of a graph.More-over,the normalized Laplacian characteristic polynomial of a graph is meaningful for eigenvalues learning.Therefore,this paper mainly studies the normalized Laplacian spectrum and character-istic polynomials of graphsIn the first chapter of this thesis,we introduce the historical development and research status of the spectral theory of graphs.Then,some concepts and notations are given.Finally,we list the results obtained in the following chapters.In the second chapter of this thesis,we study the relation between the normalized algebraic connectivity of the coalescence of two graphs and that of these two graphs by using definition of normalized algebraic connectivity.Furthermore,we investigate how the normalized algebraic connectivity behaves when the graph is perturbed by moving pendant stars,which may be used to characterize the unicyclic graph of fixed order and girth which maximizes the normalized algebraic connectivity.In the third chapter of this thesis,based on the the higher order general Randic indices for matching,we get the normalized Laplacian characteristic polynomials of unicyclic graphs and the coefficients of this polynomial.Moreover,we calculate higher order general Randic indices of some specific unicyclic graphs by matching theory,and take lollipop graph as an example to find the expression of these indices.