Research On Resistance Distance And Kirchhoff Index Of Graphs

The study of graph resistance distance and Kirchhoff index is an important branch of graph theory.It is an interdisciplinary subject in many fields,such as graph distance,graph theory,generalized inverse theory and so on.In 1993,Klein and Randi? studied the application of resistance distance in chemical graph theory from the perspective of distance function.The resistance distance and Kirchhoff index of graph play an important role in the research of graph theory,such as random walk of graph,structure analysis of graph and correlation algorithm of graph,etc,also used in complex network,system control,electrical engineering and big data science,a lot of research results are given.In this paper,the definition of local Kirchhoff index is proposed,by Laplacian block matrix and Schur complement matrix,the expression of local Kirchhoff index and the relationship between local Kirchhoff index and spanning trees of graphs are given.The relationship between local Kirchhoff index and Kirchhoff index is studied,and the relationship between local Kirchhoff index and Kirchhoff indexs under specific conditions is given.And this paper generalizes the definition of resistance distance in strongly connected balanced digraph given by R.B.Bapat.The resistance distance of strongly connected digraph is defined,by the generalized inverse of the Laplacian matrix of the graph,we give the expressions of the resistance distance and the Kirchhoff index in the Euler digraph and strongly connected graph,and the properties of the resistance distancd and the Kirchhoff index in the Euler digraph are given.