| Graph spectral theory is an important research field in graph theory,and the use of spectral theory to study the expected hitting time at any two points in the graph is an important direction in the study of modern graph theory.This paper mainly studies any two points in the graph by means of spectral theory.Expected values for random walks,degrees?Kirchhoff index and spanning trees related issues.Specific content includes:·In Chapter 1,we introduce the background and significance of the research,including the development of a representative at home and abroad regarding this aspect.Based on this research background and profound discussion,by using deep-going analysis,it fully shows the main work's necessity and innovation.·In Chapter 2,we give some necessary definitions and lemmas.·In Chapter 3,First we give a complete eigenvalue of the matrix N(Tk(G))of the graph Tk(G)in terms of those of G,and the expected hitting times of any two vertices of Tk(G)in terms of those of G is determined.Finally,as the application,based on the relationship of the expected hitting times between any two vertices of Tk(G)and G,the relation between the degree-Kirchhoff index(resp.Kemeny's constant,the number of spanning trees)of Tk(G)and G is derived.Furthermore,the resistance distance between any two vertices of Tk(G)is presented in terms of that of G.·In Chapter 4,First we give a complete eigenvalue of the matrix N(SG)of the graph SG in terms of those of G,and the expected hitting times of any two vertices of SG in terms of those of G is determined.Finally,as the application,based on the relationship of the expected hitting times between any two vertices of SG and G,the relation between the degree-Kirchhoff index(resp.Kemeny's constant,the number of spanning trees)of SG and G is derived.Furthermore,the resistance distance between any two vertices of SG is presented in terms of that of G.·In Chapter 5,we summarize the main results in this paper and give some prospects for further research in the future. |
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