The Steiner Wiener index of trees with a given segment sequence

TL;DR

This paper investigates extremal trees with a fixed segment sequence or number of segments that maximize or minimize the Steiner $k$-Wiener index, a generalization of the Wiener index in graph theory.

Contribution

It introduces the study of extremal properties of the Steiner $k$-Wiener index within classes of trees constrained by segment sequences or segment counts.

Findings

Identifies extremal trees for the Steiner $k$-Wiener index given segment sequences.

Analyzes extremal structures for trees with a fixed number of segments.

Provides characterizations of trees that optimize the Steiner $k$-Wiener index.

Abstract

The Steiner distance of vertices in a set $S$ is the minimum size of a connected subgraph that contain these vertices. The sum of the Steiner distances over all sets $S$ of cardinality $k$ is called the Steiner $k$-Wiener index and studied as the natural generalization of the famous Wiener index in chemical graph theory. In this paper we study the extremal structures, among trees with a given segment sequence, that maximize or minimize the Steiner $k$-Wiener index. The same extremal problems are also considered for trees with a given number of segments.