Exact Formula for Computing the Hyper-Wiener Index on Rows of Unit Cells of the Face-Centred Cubic Lattice

Similarly to Wiener index, hyper-Wiener index of a connected graph is a widely applied topological index measuring the compactness of the structure described by the given graph. Hyper-Wiener index is the sum of the distances plus the squares of distances between all unordered pairs of vertices of a graph. These indices are used for predicting physicochemical properties of organic compounds. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. The graphs of face-centred cubic lattice contain cube points and face centres. Using mathematical induction, closed formulae are obtained to calculate the sum of distances between pairs of cube points, between face centres and between cube points and face centres. The sum of these formulae gives the hyper-Wiener index of graphs representing face-centred cubic grid with unit cells connected in a row. In connection to integer sequences, a recurrence relation is presented based on binomial coefficients.