The Research On The Resistance Distance And Normalized Laplacian Spectral Parameters

The main task of the systems engineering is to achieve the goal of optimal design,control and management by analyzing and studying the elements of the system according to the needs for overall coordination,supplemented by modern mathematics and computer.Graph is a powerful tool to represent and study the structure of complex system.It makes the complex system to be a graph model.Thus,it is of great significance to study the structural parameters and spectral parameters of graphs.In this thesis,we mainly study the resistance distance between any two points in a graph and the spectral parameters in the line with the normalized Laplacian,such as Kirchhoff index,multiplicative degree-Kirchhoff index and expected hitting time.The details of this thesis are as follows:The first chapter introduces the research background and significance of this paper.Secondly,the research status at home and abroad is summarized.Thirdly,we introduce some well-known lemmas and proposes the key problems will be solved in this thesis.Through the analysis of the first chapter,the innovation of this thesis is fully clarified.The chapter 2 introduces the structure of linear polyomino chain and linear octagonal chain,moreover,three functions are defined.We calculate the resistance distance between any two points which are inherited from linear polyomino chain by using the principle of circuit reduction and-(35)Y transformation.The resistance distance between the new inserted points and any other points in the linear octagonal chain is determined by effective resistance sum rule and the principle of circuit reduction.The chapter 3 firstly introduces the structure of linear crossed octagonal chain and the matrix block decomposition.Secondly,the Kirchhoff(multiplicative degree-Kirchhoff,resp.)index and Wiener(Gutman,resp.)index of linear crossed octagonal chains are determined based on the Laplacian(normalized Laplacian,resp.)characteristic polynomial decomposition theorem.Finally,more surprising,one finds that the Kirchhoff(multiplicative degree-Kirchhoff,resp.)index is almost one quarter to Wiener(Gutman,resp.)index of a linear crossed octagonal chain.In Chapter 4,we first give two important lemmas.Secondly,the normalized Laplacian spectra of the two classes of parallel subdivision graphs are determined.Finally,the normalized Laplacian spectra of two classes of parallel subdivision graphs with r iterations and their corresponding spectral parameters are computed.The results obtained in this chapter generalize the previous results in [Xie et al.Applied Mathematics and Computation 2016,286: 250-256 and Guo et al.Linear Multilinear Algebral,https://doi.org/10.1080/03081087.2019.1643822].In Chapter 5,we give the normalized adjacent eigenvalues of parallel subdivision graphs.Then,the orthogonal eigenvectors corresponding to the normal adjacent eigenvalues of the parallel subdivision graphs are determined.Finally,by using the results that already obtained,the expected hitting time,resistance distance and associated invariants between any two points on the parallel subdivision graph are studied.The results obtained in this chapter generalize the previous results in [Guo et al.Linear Multilinear Algebral,https://doi.org/10.1080/03081087.2019.1643822].The chapter 6 concludes the whole thesis and presents some problems that can be further studied.This thesis include Figures 10,Tables 2,References 79.