Exact Formula Computing the Wiener Index on Rows of Unit Cells of the Diamond Cubic Grid Connected in a Row

The Wiener Index, the sum of distances between all pairs of vertices in a connected graph, is a graph invariant much studied in both mathematical and chemical literature.Topological graph indices are introduced as mathematical tools for molecule descriptions. Recently, they were computed not only for graphs representing molecules, but for other regular structured graphs including some 2 and 3 dimensional structures. The Carbon atoms in the diamond are arranged in a well defined structure. In this paper, the graph of this structure is analysed, especially, the connected part of a sequence of unit cells. The Wiener index, as the sum of the distances for every pair of atoms is computed, a closed formula depending only on the number of unit cells is proven