  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.
