The Research On The Resistance Distance And Kirchhoff Index Of Several Chain Networks

Graph theory can solve many practical problems in the real world.Graph model in graph theory,as a powerful tool to describe complex networks,has been extensively and systematically studied.Thus,it is of great significance to study the parameters of graphs.In this thesis,we mainly study the resistance distance,the Kirchhoff index,the degree Kirchhoff index and the number of spanning trees in a graph.The whole thesis is divided into six chapters,as given in the following:In the first chapter,we mainly introduce some background and significance of this paper,the research status at home and abroad and the main research content of this paper.According to the analysis of the first chapter,the innovation of this thesis is fully clarified.In the second chapter introduces the structures of linear polyomino chain,linear hexagonal chain,linear octagonal chain and linear decagonal chain,we calculate resistance distances in linear decagonal chain by using the principle of circuit reduction The resistance distance between the new inserted points and any other points in the linear decagonal chain is determined by effective resistance sum rule and the principle of circuit reduction.This chapter enriches the related research of resistance distanceIn the third chapter firstly we introduces the structures of linear octagonal-hexagonal chain,Laplacian and normalized Laplacian matrices decomposition theorem.We obtain the Laplacian and normalized Laplacian eigenvalues.Secondly,we obtain the Kirchhoff index,degree Kirchhoff index,Wiener index,Kemeny constant and number of spanning trees of linear octagonal-hexagonal chain.Finally,according to the calculation results,we find that the Kirchhoff index is almost one quarter to Wiener index of a linear octagonal-hexagonal chain.In the fourth chapter,firstly we give the structures of linear crossed octagonal-hexagonal chain.Secondly,explicit closed formula of the Kirchhoff index,degree Kirchhoff index,Wiener index and number of spanning trees of linear crossed octagonal-hexagonal chain are determined based on methods of the third chapter.Finally,we find that the Kirchhoff index is almost one quarter to Wiener index of a linear crossed octagonal-hexagonal chain.In the fifth chapter,firstly we give the structures of the Mobius phenylene chain Hn(8,6)and the cylinder phenylene chain Hn'(8,6),Secondly,explicit closed formula of the Laplacian spectrum,Kirchhoff index and number of spanning trees of their are determined based on methods of the third chapter.Finally,we find that the Kirchhoff index of the Mobius phenylene chain Hn(8,6)is smaller than the cylinder phenylene chain Hn'(8,6),and the number of spanning trees is on the contrary.In the sixth chapter,the main results of the whole thesis are summarized and we put forward the direction of further research on this basis.This thesis includes Figures 16,Tables 5,References 61.