Towards the Solution of an Extremal Problem Concerning the Wiener Polarity Index of Alkanes

TL;DR

This paper establishes sharp bounds for the Wiener polarity index of alkanes modeled as chemical trees, characterizing extremal structures and solving an open problem related to the index's maximum value.

Contribution

It provides the first precise bounds on the Wiener polarity index for chemical trees with fixed parameters and characterizes the extremal structures achieving these bounds.

Findings

Derived sharp upper and lower bounds for $W_p$

Characterized extremal chemical trees for given parameters

Solved an open problem on the maximum $W_p$ value

Abstract

The Wiener polarity index $W_p$, one of the most studied molecular structure descriptors, 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 the degree at least 3 is called a branching vertex. A segment of a chemical tree $T$ is a path-subtree $S$ whose terminal vertices have degrees different from 2 in $T$ and every internal vertex (if exists) of $S$ has degree 2 in $T$. In this paper, the best possible sharp upper and lower bounds on the Wiener polarity index $W_p$ are derived for the chemical trees of order $n$ with a given number of branching vertices or segments, and the corresponding extremal chemical trees are characterized. As a consequence of the derived results, an open problem concerning the maximal $W_p$ value of chemical trees with a fixed number of segments or branching vertices is solved.