A Novel and Efficient Method for Computing the Resistance Distance

The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph
$G$
is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to
$G$
, with each edge being replaced by a unit resistor. Hypercube
$Q_{n}$
is one of the most efficient and versatile topological structures of the interconnection networks, which received much attention over the past few years. The folded
$n$
-cube graph is obtained from hypercube
$Q_{n}$
by merging vertices of the hypercube
$Q_{n}$
that are antipodal, i.e., lie at a distance
$n$
. Folded
$n$
-cube graphs have been studied in parallel computing as a potential network topology. The folded
$n$
-cube has the same number of vertices but half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. We intend is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded
$n$
-cube by using the symmetry method and classic Kirchhoff’s equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in the Biggs-Smith graph, which shows the competency of the proposed method.