Resistance distance and Kirchhoff index of RR-vertex join and RR-edge join of two graphs

Let G=(V(G),E(G))G=(V(G),E(G)) be a graph with vertex set V(G)V(G) and edge set E(G)E(G). The RR-graph   of a graph GG, denoted by R(G)R(G), is the graph obtained from GG by adding a vertex veve and then joining veve to the end vertices of ee for each e∈E(G)e∈E(G). Let G1G1 and G2G2 be two vertex disjoint graphs. The RR-vertex join   of G1G1 and G2G2, denoted by G1〈v〉G2, is the graph obtained from R(G1)R(G1) and G2G2 by joining every vertex of V(G1)V(G1) with every vertex of V(G2)V(G2). The RR-edge join   of G1G1 and G2G2, denoted by G1〈e〉G2, is the graph obtained from R(G1)R(G1) and G2G2 by joining every vertex of I(G1)I(G1) with every vertex of V(G2)V(G2), where I(G1)I(G1) is the set of the added vertices of R(G1)R(G1). In this paper, we formulate the resistance distances and the Kirchhoff index of G1〈v〉G2 and G1〈e〉G2 respectively.