The Wiener index of a network, introduced by the chemist Harry Wiener, is the sum of distances between all pairs of nodes in the network. This index, originally used in chemical graph representations of the non-hydrogen atoms of a molecule, is considered to be a fundamental and useful network descriptor. We study the problem of constructing geometric networks on point sets in Euclidean space that minimize the Wiener index: given a set $P$ of $n$ points in $\mathbb{R}^d$, the goal is to construct a network, spanning $P$ and satisfying certain constraints, that minimizes the Wiener index among the allowable class of spanning networks. In this work, we focus mainly on spanning networks that are trees and we focus on problems in the plane ($d=2$). We show that any spanning tree that minimizes the Wiener index has non-crossing edges in the plane. Then, we use this fact to devise an $O(n^4)$-time algorithm that constructs a spanning tree of minimum Wiener index for points in convex position. We also prove that the problem of computing a spanning tree on $P$ whose Wiener index is at most $W$, while having total (Euclidean) weight at most $B$, is NP-hard. Computing a tree that minimizes the Wiener index has been studied in the area of communication networks, where it is known as the optimum communication spanning tree problem.

翻译：网络的维纳指数由化学家哈里·维纳提出，定义为网络中所有节点对之间距离的总和。该指数最初用于分子中非氢原子的化学图表示，现被认为是基础且实用的网络描述符。本文研究在欧几里得空间中构造最小化维纳指数的几何点集网络问题：给定 $\mathbb{R}^d$ 中 $n$ 个点的集合 $P$，目标是在允许的生成网络类别中构造一个覆盖 $P$ 且满足特定约束的网络，并最小化维纳指数。本研究主要关注生成树形式的网络，重点考虑平面情形（$d=2$）。我们证明，任何最小化维纳指数的生成树在平面上均无交叉边。基于此性质，我们设计了一种 $O(n^4)$ 时间复杂度的算法，用于构造凸位置点的最小维纳指数生成树。同时证明，计算 $P$ 上维纳指数不超过 $W$ 且总欧几里得权重不超过 $B$ 的生成树问题是 NP-难的。最小化维纳指数的树结构问题在通信网络领域已有研究，被称为最优通信生成树问题。