Generalizations of Wiener polarity index and terminal Wiener index

TL;DR

This paper introduces generalized versions of the Wiener polarity and terminal Wiener indices, providing efficient algorithms for their computation in trees and partial cubes, and characterizes extremal trees maximizing these indices.

Contribution

It defines new generalized indices and offers linear time algorithms for their calculation, along with extremal tree characterizations.

Findings

Linear time algorithms for computing generalized indices in trees and partial cubes.

Characterization of extremal trees maximizing the indices.

Extension of classical Wiener indices to more general graph classes.

Abstract

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertices of degree $k$. For $k = 1$, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order $n$.