Study On Laplacian Spectrum And Related Network Indexes On Extended Graphs Of Some Class Of Subdivision Graphs

With the progress of science and technology and the deepening of human cognition,complex network has attracted more and more researchers' interest.In this paper,we have studied several different composition methods related to the subdivision diagram.Chapter two is the operation of making the line graph on the basis of the multiple subdivision diagram,and constructs the iterative multiple subdivision diagram.In chapter 3,we construct heterologous or homologous multiple segmentation graphs by connecting new vertices in different ways.The fourth chapter is based on the composition of the third chapter on the heterogenetic double subdivision of the iterative operation of the heterogenetic double subdivision of the graph.We calculate the Laplace matrix eigenvalues of these graphs and some network indices related to the eigenvalues.In the first chapter,we briefly introduce the research status and background of complex networks,and emphasize the relevant background of Sierpinski graph.Then,we briefly explain how to construct the line graph,and briefly introduce how to get the adjacency eigenvalue and Laplacian eigenvalue of the graph.Finally,we introduce the concept of network consistency.In the second chapter,we construct the iterative multiple subdivision graph by using the composition method of multiple subdivision graph.Previous studies have found that network consistency is related to the Laplace spectrum of networks.On this basis,we first study the recursive formula of Laplacian eigenvalues of iterative graphs,then use the relationship between first-order and second-order consistencies and eigenvalues,and finally calculate the dominant terms of first-order and second-order consistencies of iterative multiple subdivision graphs under the condition that the network size is large enough.According to the observation results,we can draw the following conclusion: as the network size gradually increases,the stability of the iterative multiple subdivision line graph becomes stronger and stronger.In the third chapter,we consider adding connections between new vertices in a segmentation graph,which is different from the graphing of a segmentation graph in the second chapter.We further consider dividing the relationship between the newly added vertices into two types.One classifies points generated by the same edge,and the other classifies points generated by different edges.By classifying points and adding connections,we construct multiple subdivisions with heterologous or homologous origins.Firstly,we obtain the Laplacian eigenvalues of heterologous subdivision graphs by using the properties of line graphs.Then,we study the Laplace matrix eigenvalues of heterologous and homologous multi-subdivision graph.The results show that we only need to know the adjacency eigenvalues of the initial graph to obtain the Laplacian eigenvalues of the three kinds of graphs considered.In the fourth chapter,we consider to iterate the operation of heterologous multiple subdivision graph in the third chapter to obtain the iterated heterologous double subdivision graph.For the convenience of calculation,we only consider the iterative case of heterologous double subdivision graph.Firstly,we study the Laplace matrix eigenvalues of iterative graphs and give the specific expressions.From previous studies,we know that Laplacian eigenvalues have many applications.In the last section of this chapter,we study various network indicators of iterative graph,such as minimum non-zero eigenvalue,maximum eigenvalue,resistance distance and consistency analysis,through Laplace eigenvalue.In the fifth chapter,we summarize the dissertation according to the research contents.