The Resistance Distance And Kirchhoff Index Of Some Cartesian Product Graphs

The resistance distance between any two vertices in a connected graph is defined as the effective resistance between the two vertices after replacing each edge of the connected graph with a unit resistance.The sum of the resistance distances between all pairs of vertices is the Kirchhoff index of the connected grap.The resistance distance and Kirchhoff index of the graph are important topological indices of the graph,and are the structural parameters of the graph.Resistance distance and Kirchhoff index of the graph are widely used in stochastic algorithms,complex networks,electronic engineering,and chemistry.The Cartesian product graphs under different bases are different.We mainly study the Cartesian product graphs under base {(0,1),(1,1)} in this paper,the expressions of resistance distance and Kirchhoff index of Cartesian product graph of a complete graph and an arbitrary connected graph under the basis {(0,1),(1,1)} are given.We first study and give the {1}-inverse of the Laplacian matrix of a Cartesian product graph of a complete graph and any connected graph in this paper.According to the relationship between the {1}-inverse of the Laplacian matrix of the graph and the resistance distance of the graph,a resistance distance expression of a Cartesian product graph of a complete graph and an arbitrary graph is given.Then the expressions of the Kirchhoff index of the Cartesian product graph of a complete graph and an arbitrary graph are given.The results in this article generalize the conclusions of the resistance distance and Kirchhoff index expressions of the double graph.At the same time,the expressions of multiplication degree Kirchhoff index of Cartesian product graph of a complete graph and any connected graph are also given.Finally,the numerical results of the resistance distance and Kirchhoff index of the Cartesian product graphs of some special graphs and complete graphs are given.These special graph classes are tree,path,complete graph,cycle,kite graph,lollipop graph,and dumbell graph.