The hyper-Wiener Index of diamond nanowires

Carbon nanowires based on various structures have various applications. In this article, our focus is on diamond nano-wires, based on the structure of the diamond. Our goal is to characterize these nanowires by providing their hyper-Wiener index, one of the basic topological graph indices. The diamond is formed by carbon atoms in a compact structure: every atom has exactly four neighbors connected by covalent bonds according to the simplest three-dimensional polyhedron. Considering an atom in the center of a tetrahedron, its neighbors, the atoms connected to it by bonds, are located in the corners of the tetrahedron. In this article, this “diamond-grid” structure is analysed in the case that the graphs correspond to connected parts of sequences of unit cells, in this way forming nanowires of various lengths. By a combinatorial approach, a closed formula is proven for the hyper-Wiener index based on the graph-distances of every pair of carbon atoms (i.e., the minimal number of edges connecting the atoms), and this formula depends only on the size of the graph in terms of the number of unit cells included.