Towards the solution of an extremal problem concerning the Wiener polarity index of alkanes

Abstract

The Wiener polarity index Wp is a topological index that was devised by the chemist Harold Wiener for predicting the boiling points of alkanes. The index Wp 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 Wp 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 Wp value of chemical trees is provided.