Resistance Distances and Kirchhoff Indices Under Graph Operations

The resistance distance between any two vertices of a connected graph
$G$
is defined as the net effective resistance between them in the electrical network constructed from
$G$
by replacing each edge with a unit resistor. The Kirchhoff index of
$G$
is defined as the sum of resistance distances between all pairs of vertices. In this paper, two unary graph operations on
$G$
are taken into consideration, with the resulted graphs being denoted by
$RT(G)$
and
$H(G)$
. Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of
$RT(G)$
and
$H(G)$
. It turns out that resistance distances and Kirchhoff indices of
$RT(G)$
and
$H(G)$
could be expressed in terms of resistance distances and graph invariants of
$G$
. Our result generalizes the previously known result on the Kirchhoff index of
$RT(G)$
for a regular graph
$G$
to the Kirchhoff index of
$RT(G)$
for an arbitrary graph
$G$
.