Research On Topological Indices Of Several Kinds Of Chain Networks

System engineering is an interdisciplinary subject that uses comprehensive technology to solve various large-scale complex system problems.Complex system problems can often be abstracted into graph models,and the topological properties of complex system problems are proved to be closely related to the topological index of their corresponding graph models.Therefore,it is very meaningful to study the topological index of graphs.There are two types of topological indices studied in this paper.The first type is the topological index according to the vertex degree distribution of the graph,such as ABC index,GA index,AZI index,etc.The second type is the relevant topological index obtained by using spectral graph theory as a tool,such as Kirchhoff index,degreeKirchhoff index etc.The specific contents are as follows:In the first chapter,firstly,the background and significance of the research on topological indices,as well as the related conceptual terms and notations,are introduced.Secondly,the research progress and development trends in this field are described.Thirdly,the definition of topological indices relevant to the study of this paper is presented,as well as the citation.Finally,the main contents of the paper are described.In the second chapter,firstly,the network structure of six kinds of chain networks are introduced.Secondly,the edge classification of six kinds of chain networks based on vertex degree is described,respectively.Finally,specific topological index expressions for six kinds of chain networks based on degree distribution are derived,such as ABC index,GA index,Zagreb index,etc.In the third chapter,firstly,a class of octagonal ring network Tn(C8)was constructed by a single turn graph with 8 vertices through the base graph.Secondly,the resistance distance expressions between any two vertices of an octagonal ring network are discussed.Finally,the closed-form expressions of vertex degree-based topological indices,for an octagonal ring network were obtained.In the fourth chapter,firstly,the structure of linear crossed octagonal-quadrilateral chain is constructed on the basis of linear octagonal-quadrilateral chain Qn.Secondly,the(Normalized)Laplacian matrix corresponding to the linear crossed octagonalquadrilateral chain network is given.Finally,using the relevant Lemmas and theorems,the Kirchhoff index,Wiener index,Kemeny constant,degree-Kirchhoff index and the number of spanning tree expression of linear crossed octagonal-quadrilateral chain network are obtained.In the fifth chapter,the M?bius octagonal chain network Mn and cylinder octagonal chain network M'n are firstly introduced,and the Laplacian matrix corresponding to the these two kinds of chain network is given.Secondly,Kirchhoff indices and spanning trees of M?bius octagonal network and cylinder octagonal network are described,respectively,using Laplacian spectrum decomposition.Finally,through comparison,we find that the Kirchhoff index of the M?bius octagonal network is smaller than that of the cylinder octagonal network,the number of spanning tree vice versa.From the side can reflect that Mobius octagonal network is stable and better connected than the cylinder octagonal network.In addition,this chapter enriches the relevant research results of the original octagonal chain network.In the sixth chapter,we summarize the whole paper and put forward some problems that can be further studied.This thesis includes Pictures 20,Tables 6,References 78.