Study On Kirchhoff Index And Related Problems On Some Class Of Graphs

Systems engineering is a combination of quantitative and qualitative method to solve many large-scale complex system problems,and transforms these large-scale complex system problems into graph models.By studying the graph parameters of complex system,the problems of complex system can be analyzed,and the optimal design and optimal objective are obtained.For each edge in the graph G replaced by unit resistor,the resistance distance between vertices i and j is denoted asr_ij.In the study of quantitative structure activity and property,the smaller the related indexes about the resistance distance,the more stable it is.What's more,the larger the number of spanning trees,the more stable it is.This paper mainly studies the graph parameters of some class graphs,namely the Kirchhoff index,Kemeny's constant,multiplicative degree-Kirchhoff index,additive degree-Kirchhoff index,and the number of spanning trees.In the first chapter,we introduce background and significance of the research,the research status,the main problems of this paper,the lemmas and methods which can be used later.In the second chapter,according to the matrix decomposition theorem,the Laplacian matrix of linear octagonal-quadrilateral networks is decomposed into two tridiagonal matrices.The Kirchhoff index and the number of spanning trees of linear octagonal-quadrilateral networks are obtained through the structure of two matrices,Kirchhoff index and Laplacian eigenvalues of the path.In this chapter,the results enrich the related topological index of linear chains.In the third chapter,according to the structure of linear crossed phenylenes,we first investigated the Laplacian matrix and normalized Laplacian matrix of linear crossed phenylenes.Then,the Kirchhoff index,Kemeny's constant,multiplicative degree-Kirchhoff index and the number of spanning trees of linear crossed phenylenes are obtained by using matrix decomposition theorem and Vieta's theorem.Finally,it is found that the ratio of Kirchhoff index to Wiener index of linear crossed phenylenes is close to one quarter.The ratio of multiplicative degree-Kirchhoff index and Gutman index to one quarter as well.In the fourth chapter,we first construct the M(?)obius polyomino networks and Cylinder polyomino networks,and the normalized Laplacian matrices of two kinds of ring networks are obtained.Then,the Kemeny's constant,multiplicative degree-Kirchhoff index and the number of spanning trees of M(?)bius polyomino networks and Cylinder polyomino networks are obtained by using the relationship between the roots and coefficients of characteristic polynomials.Finally,it is found that the multiplicative degree-Kirchhoff index ofM(5)o(5)biuspolyomino networks is smaller than that Cylinder polyomino networks.What's more,the number of spanning trees is the opposite.In this chapter,the results from the side reflect that stability and connectivity of theM(5)o(5)biuspolyomino networks are better than the Cylinder polyomino networks.In the fifth chapter,we first introduce the graphRT(G)and graph H(G),and the vertices in the graph are classified.Then,according to the electric network and method of classification and combination,the additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index of the graphRT(G)and H(G)are obtained,respectively.Finally,the results show that the additive degree-Kirchhoff index and multiplicative degree-Kirchhoff index of the graphRT(G)andH(G)can be expressed by the graph parameters of the original graph G.To a certain extent,this result advances the existing results of resistance distance of graphsRT(G)and H(G).In the sixth chapter,the main research contents are summarized.Then,the problems that can be further studied are put forward.This thesis includes Pictures 10,Tables 9,References 77.