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 J(G) ⊙ Λ
_i=1
ⁿ
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 J(G) ⊖ Λ
_i=1
ⁿ
H
_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 J(G) ⊙ Λ
_i=1
ⁿ
H
_i
and J(G) ⊖ Λ
_i=1
ⁿ
H
_i
whenever G and H
_i
are an arbitrary graph. These results generalize the existing results.