Note on the Hyper-Wiener Index

摘要

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n_1n_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_1n_2n_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.