The Steiner Wiener index of trees with a given segment sequence

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.

Mixed-Integer Approaches to Constrained Optimum Communication Spanning Tree Problem

“…The constrained OCSTP is mainly examined in the context of the Wiener index studies (Dobrynin et al, 2001) in the mathematical chemistry. 3 Over the years, many lower bounds (including some tight bounds) are obtained for the Wiener index over different subdomains of trees, e.g., trees with the maximum degree (Fischermann et al, 2002), the given degree sequence (Zhang et al, 2008), the eccentricity sequence (Dankelmann and Dossou-Olory, 2020), or the segment sequence (Zhang et al, 2019)) and general graphs (chemical graphs (Knor et al, 2019), unicyclic graphs with the given girth (Yu and Feng, 2010), matching number (Du and Zhou, 2010), etc.). Some analytical results of this theory generalize to ODSTP (Gao and Wang, 2015;Wang and Hu, 2012) and ORSTP (Goubko and Kuznetsov, 2020).…”