Wiener index on rows of unit cells of the face-centred cubic lattice

The Wiener index of a connected graph, known as the `sum of distances', is the first topological index used in chemistry to sum the distances between all unordered pairs of vertices of a graph. The Wiener index, sometimes called the Wiener number, is one of the indices associated with a molecular graph that correlates physical and chemical properties of the molecule, and has been studied for various kinds of graphs. In this paper, the graphs of lines of unit cells of the face-centred cubic lattice are investigated. This lattice is one of the simplest, the most symmetric and the most usual, cubic crystal lattices. Its graphs contain face centres of the unit cells and other vertices, called cube vertices. Closed formulae are obtained to calculate the sum of shortest distances between pairs of cube vertices, between cube vertices and face centres and between pairs of face centres. Based on these formulae, their sum, the Wiener index of a face-centred cubic lattice with unit cells connected in a row graph, is computed.