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

TL;DR

This paper derives formulas for resistance distance and Kirchhoff index in two types of corona graphs based on subdivision graphs, generalizing previous results to arbitrary graphs.

Contribution

It provides the first general formulas for resistance distance and Kirchhoff index in corona-vertex and corona-edge subdivision graphs for arbitrary graphs.

Findings

Formulas for resistance distance in $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$.

Formulas for Kirchhoff index in these corona graphs.

Generalization of previous specific cases.

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 $G_{1}\diamondsuit G_{2}$ 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 $G_{1}\diamondsuit G_{2}$ and $G_{1}\star G_{2}$ when $G_{1}$ and $G_{2}$ are arbitrary graphs. These results generalize them in $\cite{PL}$.