Bounding the $k$-Steiner Wiener and Wiener-Type Indices of Trees in Terms of Eccentric Sequence

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 the minimum 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.