Bounding the $k$-Steiner Wiener and Wiener-type indices of trees in terms of eccentric sequence

TL;DR

This paper characterizes trees with given eccentric sequences that minimize various distance-based indices, including Wiener and Steiner indices, unifying bounds and correcting previous errors in the literature.

Contribution

It extends previous results on Wiener index minimization to a broad class of Wiener-type indices and the $k$-Steiner Wiener index, providing new bounds and identifying extremal trees.

Findings

Identifies trees minimizing Wiener-type indices for given eccentric sequences.

Provides sharp lower bounds for the $k$-Steiner Wiener index based on order and diameter.

Corrects a known error regarding extremal trees for the $k$-Steiner Wiener index.

Abstract

The eccentric sequence of a connected graph $G$ is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the Harary index, the generalised Wiener index $W^{\lambda}$ for $\lambda>0$ and $\lambda <0$, and the reciprocal complementary Wiener index. Our results imply and unify known bounds on these Wiener-type indices for trees of given order and diameter. We also present similar results for the $k$-Steiner Wiener index of trees with a given eccentric sequence. The Steiner distance of a set $A\subseteq V(G)$ is theminimum number of edges in a subtree of $G$ whose vertex set contains $A$, and the $k$-Steiner Wiener index is the sum of distances of all $k$-element subsets of $V(G)$. As a corollary, we obtain a sharp lower bound on the $k$-Steiner Wiener index of trees with given order and diameter, and determine in which cases the extremal tree is unique, thereby correcting an error in the literature.