Resistance Distance and Kirchhoff Index of Generalized Subdivision-Vertex and Subdivision-Edge corona for Graphs

Let G be a connected graph. 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 such new vertices is denoted by I(G). The generalized subdivision-vertex corona of G and H-i for i = 1, 2, ..., n, denoted by S(G)circle dot boolean AND(n)(i=1) H-i, is the graph obtained from S(G) and H-i by joining the ith vertex of V(G) to every vertex in H-i The generalized subdivision-edge corona of G and H-i for i = 1, 2, ..., m, denoted by S(G) circle minus boolean AND H-m(i=1)i, is the graph obtained from S(G) and H-i by joining the ith vertex of I(G) to every vertex in H-i. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of S(G)circle dot boolean AND H-n(i=1)i and S(G)circle minus boolean AND(m)(i=1)Hi whenever G and H-i are an arbitrary graph. These results generalize the existing result.