Delaunay decompositions minimizing energy of weighted toroidal graphs

TL;DR

This paper demonstrates that for weighted toroidal graphs, the energy-minimizing harmonic map corresponds precisely to a weighted Delaunay decomposition, revealing a deep geometric structure linking energy minimization and Delaunay tessellations.

Contribution

It establishes a novel connection between energy minimization of harmonic maps and weighted Delaunay decompositions for toroidal graphs.

Findings

Energy-minimizing harmonic maps induce weighted Delaunay decompositions.

Optimal Euclidean structures are characterized by Delaunay properties.

The approach links graph energy optimization with geometric tessellations.

Abstract

Given a weighted toroidal graph, each realization to a Euclidean torus is associated with the Dirichlet energy. By minimizing the energy over all possible Euclidean structures and over all realizations within a fixed homotopy class, one obtains a harmonic map into an optimal Euclidean torus. We show that only with this optimal Euclidean structure, the harmonic map and the edge weights are induced from a weighted Delaunay decomposition.