Resistance distance and Kirchhoff index of the splitting-joins of two graphs

Abstract

Let $G$ be a graph. The splitting graph $SP(G)$ of $G$ is the graph received from $G$ by putting a new vertex $w'$ for each $w\in V(G)$ and joining $w'$ to all vertices of $G$ adjacent to $w$. Let $S_G$ be the set of such new vertices of the splitting graph $SP(G)$. Let $G_1$ and $G_2$ be two simple connected graphs, the splitting $V$-vertex join graph is obtained by taking one copy of $SP(G_1)$ and joining each vertex in $V_{G_1}$ to each vertex in $V_{G_2}$, denoted by $G_1\veebar G_2$. The splitting $S$-vertex join of $G_1$ and $G_2$, denoted by $G_1 \barwedge G_2$, is a graph obtained from $SP(G_1)$ and $G_2$ by joining each vertex in $S_{G_1}$ to each vertex in $V_{G_2}$. In this paper, we calculate the resistance distance and Kirchhoff index of $G_1\veebar G_2$ and $G_1\barwedge G_2$ for regular graphs $G_1$ and $G_2$, respectively.