Resistance Distance of Three Classes of Join Graphs

Abstract

The subdivision graph $S(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. The set of inserted vertices of $S(G)$ is denoted by $I(G)$. Let $G_1$ and $G_2$ be two vertex-disjoint graphs. The subdivision-edge-vertex join of $G_1$ and $G_2$, denoted by $G_1\odot G_2$, is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining every vertex in $I(G_1)$ to every vertex in $V(G_2)$. The subdivision-edge-edge join of $G_1$ and $G_2$, denoted by $G_1\ominus G_2$, is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining every vertex in $I(G_1)$ to every vertex in $I(G_2)$. The subdivision-vertex-edge join of $G_1$ and $G_2$, denoted by $G_1\odot G_2$, is the graph obtained from $S(G_1)$ and $S(G_2)$ by joining every vertex in $V(G_1)$ to every vertex in $I(G_2)$. In this paper, we obtain the formulas for resistance distance of $G_1\odot G_2$, $G_1\ominus G_2$, and $G_1\odot G_2$.