A Novel and Efficient Method for Computing the Resistance Distance

Abstract

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 structure 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 graph have been studied in parallel computing as a potential network topology. The folded n-cube has the same number of vertices but the half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. Our intention 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 Biggs-Smith graph, which shows the competency of the proposed method.

Transport collapse in dynamically evolving networks

“…The topology of lattice-based networks is similar to the one of hypercubic graphs Q n , but with an increased number of nodes per edge. We stress that Q n are commonly used and appreciated in data sciences for both their versatile topology and efficiency in distributing numerical data between nodes in interconnection networks [53][54][55]. They are efficient structures for transport, which are also resilient to attacks [55,56].…”