Research On Resistance Distance And Kirchhoff Index Of Some Graphs

In 1993,on the basis of network theory,D.J.Klein and M.Randi?defined a new distance function named resistance distance r_ij.The resistance distance r_ij constructed from N by replacing each edge of G with a unit resistor.The resistance distance of G is viewed as the effective resistance between vertices i and j in the corresponding network.The sum of resistance distances between all pairs of vertices of G is the Kirchhoff index.The concept of the resistance distance is widely used in Complex networks,Chemistry,Random walk and so on.In the first chapter of this dissertation,we introduce the historical background and the research status of the resistance distance and the Kirchhoff index.In chapter 2,we introduce some knowledge about graph theory and algebra theory.In chapter 3 and 4,we mainly study the resistance distance and the Kirchhoff index of some graphs,the main results is as follows:Firstly,in this paper,we get the formulas for the resistance distance and the Kirchhoff index of the RT?G?by using the Laplacian matrix,Genralized inverse and Group inverse,and calculate the Laplace energy and the number of the spanning tree.Secondly,in this paper,we get the normalized Laplacian eigenvalues and the Degree Kirchhoff index of theR^k?G?by using the definition of the normalized Laplacian matrix,and calculate the Laplace energy and the number of the spanning tree.In addition,in this paper,we get the formula of the Kirchhoff index of the G₁'G₂ and obtain the inequality of the add Degree Kirchhoff index and the multiplication Degree Kirchhoff index by using the polynomial of the Laplacian matrix.