The Generalized Characteristic Polynomials Of Composite Graph And Its Applications

For a given graph G with adjacency matrix A(G),the generalized characteristic polynomial of G is defined to be?_G(x,t)=det(xI-(A(G)-tD(G))),where D(G)is the diagonal degree matrix of G.The generalized characteristic polynomial not only generalizes the characteristic polynomials of adjacency,Laplacian,signless Laplacian and normalized Laplacian matrices,but closely relates to the Bartholdi Zeta function.But up to now,the results about it is relative less.In this paper,we mainly discuss the generalized characteristic polynomial of several kinds of composite graphs and its application.In Chapter 1,we first give some basic concepts and notation which are needed in this thesis,then simply introduce the background of the topic.In Chapter 2,we determine the generalized characteristic polynomials of four kinds of composite graphs(triangulation-vertex join,subdivision-vertex join,subdivision-edge join and the generalized join)in terms of parameters of its factor graphs.As applications of the results obtained in Chapter2,we obtain the explicit expressions of the Bartholdi Zeta function for the first three kinds of composite graphs.In the last Chapter of this thesis,as applications,we compute the number of spanning trees and the Kirchhoff Index of several kinds of graphs.At the same time,we give ways to construct graphs which are cospectral for the adjacency,Laplacian,signless Laplacian,normalized Laplacian matrices at the same time.