On Combination Structural Properties Of Self-similar Complex Networks

In this paper,we investigate the combination structural properties of some class of self-similar complex networks.Extensive studies indicate that the combination structural properties of a network are closely related to its topo-logical characteristics because it can be used to reflect or describe some new properties of real-life world network or the impacts taken by them.This thesis consists of six chapters.In the first chapter,we summarize some recent investigation progresses of complex network and give some neces-sary definition.In chapter two,we study the combination structural properties of Sierpinski-like self-similar networks.First we consider the dimer-monomers model of the Hanoi graph,and the high accuracy asymptotic growth constant for it’s dimer-monomers model is obtained.Second we study the independent sets of the Hanoi graph and 3-prism network.In the last part of chapter two,we determine the Wiener Polarity index of some Sierpinski-like networks.In chapter three,we study the combination structural properties of two families of scale-free self-similar networks.We first give the Tutte polynomial of the scale-free networks by applying the subgraph-decomposition method,respectively.As the application of Tutte polynomial,some graph invariants of the two networks are determined,including the number of spanning trees,the number of acyclic orientations,etc.In the last part of chapter three,we obtain the Wiener Polarity index of the scale-free networks.In chapter four,we investigate analytically the critical characteristics of the recursive corona small-world network,including degree distribution,clus-tering coefficient,average path length,and Detour index,as well as Kirchhoff index.Furthermore,a compact analytical expression of Tutte polynomial of the recursive corona small-world network is derived.In chapter five,we first study some Kirchhoff-type invariants for the gen-eral diamond operation networks St(G)which is defined as replace each edge in the underline graph G by t copes of paths of length two.We derive formulae for the Kirchhoff index,the additive degree Kirchhoff index and multiplicative degree Kirchhoff index of St(G).Furthermore,we also give the spectra of the normalized Laplacian of St(G).As applications,the multiplicative degree-Kirchhoff index,the Kemeny’s constant and the number of spanning trees of the iterative network Stk(G)are derived.In chapter six,we study the combination structural properties of a class of non-uniform polycyclic chains.We get an explicit formulae for the Tutte poly-nomial of the non-uniform polycyclic chains,which generalizes the previous result of uniform polycyclic chains.In the last chapter,we summarize the main work of this thesis and raises several prospects for further study in the future.