Abstract
The Wiener polarity index W-p is a topological index that was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index W-p for chemical trees (chemical graphs representing alkanes) is defined as the number of unordered pairs of vertices at distance 3. A vertex of a chemical tree with degree at least 3 is called a branching vertex. A segment of a chemical tree T is a nontrivial path S whose end-vertices have degrees different from 2 in T and every other vertex (if exists) of S has degree 2 in T . In this paper, sharp upper and lower bounds on the Wiener polarity index W-p are derived for the chemical trees of a fixed order and with a given number of branching vertices or segments, and for every such bound, a class of trees attaining that bound is obtained. As a consequence of the derived results, a vital step towards the complete solution of an existing open problem concerning the maximum W-p value of chemical trees is provided. (C) 2021 Elsevier Ltd. All rights reserved.