Wiener Index on Lines of Unit Cells of the Body-Centered Cubic Grid

Abstract

The Wiener Index of a graph, known as the sum of distances of a connected graph, is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. In this paper, the lines of unit cells of the body-centered cubic grid are used. These graphs contain center points of the unit cells and other vertices, called border vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of border vertices, between border vertices and centers and between pairs of centers. Based on these formulae, their sum, the Wiener Index of body-centered cubic grid with unit cells connected in a row graph is computed. Some relationships between formulae and integer sequences are also presented.