The Generalized Characteristic Polynomials Of Some Graphs And Its Related Issues

Spectral graph theory is one of the important fields of graph theory,which mainly re-search directions include spectral characteristics of graphs,graph characteristics of spectra and graph related issues,and the aim to reflect the spectral properties of graphs through the eigenvalues of graph matrix and establish the relationship between the topological structure of graphs and its graph.Let G be a graph and A(G)and D(G)be the adjacen-cy matrix and degree diagonal matrix,respectively.We define M(G)=A(G)-tD(G)is the generalized matrix of G,and further the generalized characteristic polynomial is defined as ΦG(λ;t)=det(λIn-M(G))=det(λIn-(A(G)-tD(G))).In this paper,the research on generalized characteristic polynomials of graphs divided into the following aspects:the properties of generalized characteristic polynomials,the generalized char-acteristic polynomial of some generalized graphs,such as the generalized corona graph(?)Hi,the generalized edge corona graph G[Hi]1m the modified generalized edge corona graph G[Hi]1m the generalized of subdivision Q-corona graphs and subdivision Q-neighborhood corona graphs.This thesis is organized as follows:In Chapter one,we introduce the research background,research trends and related problems of the generalized characteristic polynomial of graphs,and also list introduces related concepts of generalized characteristic polynomial are introduced.In Chapter two,we consider the generalized characteristic polynomial and then present all main results of the paper,the generalized characteristic polynomial of com-plement graph G of G,at the same time,the generalized characteristic polynomial of of complement graph GUH and the generalized characteristic polynomial of join G▽H of G and H.In Chapter three,we present the generalized characteristic polynomials of the gener-alized corona graph(?)Hi,the generalized edge corona graph G[Hi]1m the modified generalized edge corona graph,meanwhile,their Φ-spectra are obtained.As applications,the number of the spanning trees and Kirchhoff index are obtained.In Chapter four,we give the generalized characteristic polynomials of the subdivision Q-vertex corona and the subdivision Q-edge corona and its Φ-spectrum,at the same time,we got it A-spectrum、L-spectrum、Q-spectrum and(?)-spectrum.As applications,the energy,the number of the spanning trees,Kirchhoff index,Degree-Kirchhoff index and Kemeny constant are obtained.In Chapter five,we obtain the generalized characteristic polynomials and Φ-spectrum of the subdivision Q-neighborhood vertex corona and the subdivision Q-neighborhood edge corona,at the same time,we got it A-spectrum、L-spectrum、Q-spectrum and C-spectrum.As applications,the energy,the Laplacian-energy-Like,the incidence energy and its Φ-cospectra are obtained.