Resistance Distance and Kirchhoff Index of the Corona-Vertex and the Corona-Edge of Subdivision Graph

Abstract

The resistance distance is widely used in random walk, electronic engineering, and complex networks. One of the main topics in the study of the resistance distance is the computation problem. The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$ . Two classes of new corona graphs, the corona-vertex of the subdivision graph $G_{1}\diamondsuit G_{2}$ and the corona-edge of the subdivision graph $G_{1}\star G_{2}$ , were defined by Lu and Miao. The adjacency spectrum and the signless Laplacian spectrum of the two new graph operations were computed when $G_{1}$ is an arbitrary graph and $G_{2}$ is an $r_{2}$ -regular graph. In this paper, we give the closed-form formulas for the resistance distance and Kirchhoff index of $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ in terms of the resistance distance and Kirchhoff index of the factor graphs.