Note of the hyper-Wiener index

Abstract

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n1n2, over all pairs υ,ν of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects υ and ν. We examine a slight modification WWW of the hyper-Wiener index defined as the sum of the products n1n2n3, over all pairs υ,ν of vertices of T, where n3 is the number of vertices of T, lying between υ and ν. 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.