Analysis For Network Indices And N-expansions Properties

In this paper,we mainly study the network indices and the properties of N-expansions.This paper is divided into five parts.In the first part,the development process and research status of complex networks and weighted networks are introduced,and the basic concepts and network indices of complex networks and weighted networks are given.Furthermore,we also state the basic concepts and properties of continued fraction.In the second part,we construct a kind of k-th weighted iterated quadrilateral graphs,τk(G).For any generation k,we deduce the spectrum of τk(G)by studying the their topologies,and they are used to get the graphs’ multiplicative Kirchhoff index and Kemeny’s constant.It reveals that multiplicative Kirchhoff index and Kemeny’s constant of τk(G)are related to the number of iteration generation k,the weight factor and the topology of initial graph.In the third part,a kind of k-th weighted iterated q-triangulations of graphs,τqk(G),is discussed to explore the influence of the complexity factor q on the normalized Laplacian spectrum and its related performance indicators of each generation of the weighted graph.By analyzing the spectrum connection of two graphs,τqk-1(G)and τqk(G),the exact expression of the normalized Laplacian spectrum of τqk(G)related to q and k is obtained,then they are applied to argue that some relevant invariants of τqk(G)are relevant to number of iteration generation k,weight factor,complexity factor q and topology of initial graph.In the fourth part,we continue to explore a class of k-th iterated joint graphs of regular graph,τk(G,G’),where G’ is regular.Consider the networks unweighted,the Laplacian spectrum relation between τk-1(G,G’)and τk(G,G’)is obtained.And the relevant algorithms are also given to show the Kirchhoff index,Kemeny’s constant,and the number of spanning trees for τk(G,G’)at every generation k.In the fifth part,we consider a class of continued fraction expansions:the socalled N-expansions with a finite digit set.We prove that for all integers N≥ 2,at least one explicit interval Iα*is determined,such that for all α∈Iα*the entropy h(Tα)of the underlying Gauss-map Ta is equal,besides that,we also show the entropy value.Furthermore,we also discuss the number of interval Iα*for integer N≥ 2.In order to show that the entropy is constant on such interval,we obtain the underlying planar natural extension of the map Tα,the Tα-invariant measure,ergodicity,and we show that for any two α1,α2 ∈Iα*,the natural extensions are metrically isomorphic.