The Normalized Laplacian Spectrum Of N-Polygon Graphs And Its Applications

Given an arbitrary connected G,the n-polygon graph ?n(G)is obtained by adding a path with length n(n?2)to each edge of graph G,and the iterated npolygon graphs ?ng(G)(g?0),is obtained from the iteration ?ng(G)=?n(?ng-1(G),with initial condition Tn0(G)=G.In this paper,a method for calculating the eigenvalues of normalized Laplacian matrix for graph ?n(G)is presented if the eigenvalues of normalized Laplacian matrix for graph G is given firstly.And the normalized Laplacian spectrums for the graph ?n(G)and the graphs ?ng(G)(g>0)can also be derived.Firstly,it summarizes the research significance,historical background and development status of this topic.Some basic knowledge,general notations,classical results and important lemmas are given.Three important lemmas play a key role in the subsequent proof.Secondly,the relation between the normalized Laplacian spectrum of arbitrary undirected connected graph G and the normalized Laplacian spectrum of polygonal iterative graph ?n(G)is proved.Then the method of obtaining the normalized Laplacian spectrum of polygon iteration graph ?n(G)is proved.Then,as an application,we use the method of obtaining the normalized Laplace spectrum of polygonal iteration graph ?n(G),and obtain the normalized Laplace spectrum of g times iteration graph ?ng(G)(g>0)of graph G,and Some indexes to measure the efficiency of web search are calculated,such as multiplicative degree-Kirchhoff index,Kemeny's constant and the number of spanning trees.Finally,the main content of this paper is summarized,and our follow-up research work and prospects for the future are introduced.