Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs

Abstract

The central graph $C(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into each edge of $G$ exactly once and joining all the non-adjacent vertices in $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The central vertex join of $G_1$ and $G_2$ is the graph $ G_1\dot{\vee} G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex of $G_1$ with every vertex of $G_2$. The central edge join of $G_1$ and $G_2$ is the graph $ G_1\veebar G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex corresponding to the edges of $G_1$ with every vertex of $G_2$. In this article, we obtain formulae for the resistance distance and Kirchhoff index of $G_1\dot{\vee} G_2$ and $ G_1\veebar G_2$. In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph.