Study Of Generalized Characteristic Polynomials And Resistance Distance Of Graphs

Spectral graph theory and resistance distance is an important branch of graph theory,which is widely used in computer science,communication,physics,quantum chemistry and other fileds.Let M be a n×n matrix,the characteristic polynomial of the matrix M is defined asψ（M,x）=det（xI_n-M）,and the matrix I_n is the matrix which has the same dimensions with the matrix M,the spectrum of a graph is the eigenvalue of characteristic polynomials of graphs including their multiplicity.Let G be a graph on n vertex,and A,D,L,Q,%L be the adjacency matrix,degree matrix,Laplacian matrix,signless Laplacian matrix,normalized Laplacian matrix,the generalized characteristic polynomial of G is defined asφ（x,t）=det（xI_n-（A-tD））.And when t=0,t=1,t=-1 and t=-x+1,we haveψ（A,x）=φ（x,0）,ψ（L,x）=（-1）^|V|φ（-x,1）,ψ（Q,x）=φ（x,-1）,ψ（%L,x）=（-1）^|V|φ（0,-x+1）/det（D）.The generalized characteristic polynomial assemble the adjacency characteristic polynomial,Laplacian characteristic polynomial,signless Laplacian characteristic polynomial and normalized Laplacian characteristic polynomial,since which can reduce the computational workload of the spectrum.Resistance distance is defined as the effective electrical resistance between two vertices if each edge of G is replaced by a unit resistor and then constructed a corresponding electrical network.Resistance distance is a measure of distance defined in a graph,and also a important invariant of a graph,the sum of the resistance distance between all the vertices in the graph is the Kirchhoff index of the graph.In this thesis,some classes of graphs are studied:the edge-subdivision-vertex corona G₁∨G₂,edge-subdivision-edge corona G₁?G₂,subdivision-vertex-corona join G₁?G₂,subdivision-edge-corona joinG₁?G₂ obtained byG₁ andG₂.The generalized characteristic polynomial of a graph is obtained through the generalized matrix of a graph,and some class of spectral of edge-subdivision-vertex corona G₁∨G₂ and edge-subdivision-edge coronaG₁?G₂ are obtained,which solve the problem of getting the spectrum of these complex graphs.At the same time,the resistance distance of subdivision-vertex-corona joinG₁?G₂and subdivision-edge-corona join G₁?G₂ is proved.As application,we calculate the number of spanning trees,Kirchhoff index and energy of the graph.In this thesis,we get the main results as follows:（1）Calculate and prove the generalized characteristic polynomial,adjacency characteristic polynomial,Laplacian characteristic polynomial,signless Laplacian characteristic polynomial,normalized Laplacian characteristic polynomial and some spectrum of the edge-subdivision-vertex corona G₁∨G₂ and edge-subdivision-edge corona G₁?G₂.（2）The several kinds of energy,Kirchhoff index and spanning tree and generalized cospectra graphs of edge-subdivision-vertex corona G₁∨G₂ and edge-subdivision-edge corona G₁?G₂ are obtained.（3）Calculate and prove the resistance distance between any two vertices of the subdivision-vertex-corona joinG₁?G₂,subdivision-edge-corona joinG₁?G₂,and obtained the Kirchhoff index.And calculate the Laplacian generalized inverse matrix and resistance distance matrix of the subdivision-vertex-corona join P₃?P₂ and subdivision-edge-corona join C₃?P₂.