Computation of resistance distance with Kirchhoff index of body centered cubic structure

Abstract

The two-point effective resistance of an electrical network is a classical problem in theory of electrical networks. In a graph G the resistance distance among two vertices is an effective resistance between the respective vertices in the extracted electrical network from a graph G by setting every edge of graph G with a unit resistor. The sum of resistance distance between all pairs of vertices of graph G is the Kirchhoff index. Body centered cubic unit cells are made of atoms arranged in a cube, with one atom at each corner and another atom at the center. Many chemists and mathematicians have studied the body centered cubic structure (BCC) due to its atom arrangement. Utilizing some techniques from electrical networks theory, we compute the effective resistance among every pair of vertices of body centered cubic structure (BCC). We also apply our results to derive the formula for the Kirchhoff index.