Note on the hyper-Wiener index

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n(1)n(2), over all pairs u, v of vertices of T, where n(1) and n(2) are the number of vertices of T, lying on the two sides of the path which connects u and v. We examine a slight modification WWW of the hyper-Wiener index, defined as the sum of the products n(1)n(2)n(3), over all pairs u, v of vertices of T, where n(3) is the number of vertices of T, lying between u and v. It is found that WWW correlates significantly better with various physico-chemical properties of alkanes than WW. Lower and upper bounds for WWW, and an approximate relation between WWW and WW are obtained.