Study On The Correlation Spectrum Of Distance Matrix Of Graphs And Its Application

Spectral graph theory,as an important research direction of graph theory,mainly studies the topological structure and algebraic properties of graphs through all kinds of matrices(adjacency matrix,distance matrix,distance Laplacian matrix,distance signless Laplacian matrix,etc.),eigenvalues and eigenvectors of graphs,which are widely used in computer,complex systems,chemistry,physics and other disciplines.Let G be a simple connected graph with vertex set V(G)and edge set E(G).The distance matrix is defined as D(G)=(dvivj)n×n,where dvivj represents the distance between vertices vi and vj,that is,the length of the shortest path.The transmission degree Tr(vi)of a vertex vi is defined as the sum of the distances from vertex vi to other vertices in G,that is,Tr(vi)=?vj?V(G)dG(vi,vj),the diagonal matrix of transmission degree of G is denoted by Tr(G).The distance Laplacian matrix and distance signless Laplacian matrix are defined as L(G)=Tr(G-D(G)and Q(G)=Tr(G)+D(G),respectively.The set of eigenvalues and their corresponding multiplicities of the matrix of G is called the spectrum of G.The spectrum of V(G),L(G)and Q(G)are denoted by D-spectrum,L-spectrum and Q-spectrum,respectively.The generalized distance matrix is defined as D?(G)=?Tr(G)+(1-?)D(G),0???1.Therefore,the distance matrix,the distance Laplacian matrix and the distance signless Laplacian matrix can be linked together by a parameter ?,which greatly reduces the workload of spectrum calculation.The set of eigenvalues and their corresponding multiplicities of D?(G)is called the generalized distance spectrum,written D?-spectrum.The maximum eigenvalue p(G)of D?(G)is called the generalized distance spectral radius of G.In this thesis,we mainly calculate the D-spectrum,L-spectrum,Q-spectrum and D?-spectrum of several kinds of composite graphs.The distance(signless)Laplacian spectral energy of some composite graphs is calculated and a kind of distance(signless)Laplacian integral graphs is constructed.In addition,we present some new upper and lower bounds on the generalized distance spectral radius of G,based on other graph-theoretic parameters,and characterize the extremal graphs.Finally,the lower bounds of the generalized distance spectral radius of line graph L(G)of graph G are obtained.As an application,the local dimension centrality index based on distance is proposed,and the application of distance matrix in the robustness of complex networks is studied.The main results are as follows:1.Calculate the L-spectrum and Q-spectrum of the cluster graph G{Km},double graph D2G,join of regular graphs G1?G2,G1?(G2UG3);Calculate the D-spectrum,L-spectrum and Q-spectrum of the subdivision-edge join G1?G2,subdivision-vertex join G1?G2 and subdivision-(vertex-edge)join ?.As a corollary,obtain the L-spectrum and Q-spectrum of the complete bipartite graph Km,n;As an application,Calculate the distance(signless)Laplacian energy of double graph of the transmission regular graph.2.Calculate the D?-spectrum of the Indu-B ala product G1?G2,lexicographic product G?H,cubic lattice graph and C4 nanotori graph Tk,m,C4;As a corollary,obtain the L-spectrum and Q-spectrum of the G1?G2;As an application,obtain a class of distance(signless)Laplacian integral graph Kn?Kn+1 and Calculate the distance(signless)Laplacian energy of Kn?Kn+1.In addition,obtain some new upper and lower bounds on the generalized distance spectral radius of G,based on other graph-theoretic parameters,and characterize the extremal graphs.Finally,obtain the lower bounds of the generalized distance spectral radius of line graph L(G)of graph G.3.As the application of distance matrix in complex network,propose a local dimension centrality index(LD)based on distance r.The attack effects of DC,BC,CC,KS and LD on WS small world network,BA scale-free network and protein network are compared.The results show that the attack effect of LD index on network is better than other indexes,and the robustness of these three networks is analyzed.