Research On Resistance Distance Of Some Graph Operations

The research of distance of graph is a very important research field,D.J.Klein and M.Randi(?) defined resistance distance based on network theory in 1993,resistance distance is an explanation for the concept of vis-à-vis resistance in electrical networks,it reflects a communication distance.The concept of resistance distance is widely used in random walk of graph,structure analysis of graph and correlation algorithm of graph,etc.The sum of resistance distance of all pairs in graph is Kirchhoff index of the graph.The research of resistance distance and Kirchhoff index of graph operations is also an important research topic.The calculation formulas of resistance distance and Kirchhoff index of many graph operations are also obtained.The history and current research status of resistance distance are introduced in the part of introduction,the basic knowledge of graph theory and matrix theory is given in the part of preliminaries,the resistance distance of power hypergraph is defined and researched in Chapter 3,the resistance distance and Kirchhoff index of power hyper-vertex join graph are researched in Chapter 4,by using the generalized inverse of Laplacian matrix,the calculation formulas of the resistance distance and Kirchhoff index of power hypergraph and power hyper-vertex join graph are given,and the relationship between the 2 operations and known graph operations is obtained,the relationship between resistance distance and Kirchhoff index and the number of walks of original graphs under certain conditions is researched.The calculation formulas of resistance distance and Kirchhoff index of Q-vertex join graph and Q-edge join graph are given in Chapter 5,and the relationship between resistance distance and Kirchhoff index and the number of walks of original graph is given.