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 \(G_1 \odot G_2\).