Resistance distance and Kirchhoff index in the corona-vertex and the corona $-$ edge of subdivision graph

Abstract

The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. In $\cite{PL}$, two classes of new corona graphs, the corona-vertex of the subdivision graph $G_1♢G_2$ and corona-edge of the subdivision graph $G_1\star G_2$ were defined. The adjacency spectrum and the signless Laplacian spectrum of the two new graphs were computed when $G_1$ is an arbitrary graph and $G_2$ is an $r$-regular graph. In this paper, we give the formulate of the resistance distance and the Kirchhoff index in $G_1♢G_2$ and $G_1\star G_2$ when $G_1$ and $G_2$ are arbitrary graphs. These results generalize them in $\cite{PL}$.