Study On Laplacian Spectrum And Related Network Indexes On Four Classes Of Networks

With the advancement of science and technology and the deepening of human cognition,complex networks have increasingly attracted researchers’interest.In this paper,we study several types of networks the weighted scale-free networks,the vertex-vertex graphs,the partition network by substitution rule and the weighted directed tree networks.We calculate Laplacian spectrum and some index of several types of networks.In the first chapter,we introduce the relevant research background and research status quo of complex networks,the current research situation of weighted network,some indexes of network indicators and the primary contents for this thesis.In the second chapter,we introduce the weighted scale-free networks by iterative construction,and each edge is weighted in a certain way.Then,we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations.Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network.Then,we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix.Finally,we show some applications of the Laplacian spectrum.In the third chapter,we define the one vertex–vertex graph,and we found that we could not directly obtain the eigenvalues of the Laplace matrix of the vertex-vertex graph by the relationship between the eigenvalues and the eigenvectors.Then,we introduce the preliminaries theorem and we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix,and we obtain the Laplacian spectrum for vertex–vertex graph.Based on the review of these literatures which have found that the recursive relationship of the eigenvalues of Laplacian matrix at two successive generations satisfies the following two equations.One is a quadratic equation,the other is a cubic equation.In this paper,one linear equation,which shows the recursive relationship of the eigenvalues of Laplacian matrix at two successive generations of the vertex–vertex graph.The new findings could better improve the theory of eigenvalues of Laplacian matrix.Finally,we show the applications of the Laplacian spectrum.In the forth chapter,according to the symmetry feature of the self-similar network,the adjacency matrix,strength matrix and transfer matrix of the network can be obtained by using the property of network symmetry and weighted with certain rules.Next,we can deduce the the recursive relationship of its eigenvalues at two successive generations of transition weighted matrix.Then,we obtain eigenvalues of Laplacian matrix by the two successive generations of transition weighted matrix.Finally,we calculate the accurate expression of eigentime identity and Kirchhoff index by the spectrum of Laplacian matrix.In the fifth chapter,we further study the average trapping time of the weighted directed treelike network constructed by an iterative way.Then,we introduce our model inspired by trade network,each edgee_ij in undirected network is replaced by two directed edges with weightsw_ij andw_ji.Then,the trap located at central node,we calculate the weighted directed trapping time and the average weighted directed trapping time.Remarkably,the weighted directed trapping time has different formulas for even generations and odd generations.Finally,we analyse different cases for weight factors of weighted directed treelike network.