Spectrally Optimized Pointset Configurations

TL;DR

This paper introduces a new spectral optimization criterion for pointset configurations, analyzing their properties through graph spectral invariants and identifying extremal configurations like the regular simplex and triangular lattice.

Contribution

It proposes a novel spectral-based optimality criterion for pointsets, connecting geometric arrangements with spectral graph theory and providing both theoretical and computational insights.

Findings

Regular simplex extremizes several spectral invariants on the sphere.

Triangular lattice is often extremal for spectral functions on flat tori.

Spectral properties relate to applications in solar cell design and swarming models.

Abstract

The search for optimal configurations of pointsets, the most notable examples being the problems of Kepler and Thompson, have an extremely rich history with diverse applications in physics, chemistry, communication theory, and scientific computing. In this paper, we introduce and study a new optimality criteria for pointset configurations. Namely, we consider a certain weighted graph associated with a pointset configuration and seek configurations which minimize certain spectral properties of the adjacency matrix or graph Laplacian defined on this graph, subject to geometric constraints on the pointset configuration. This problem can be motivated by solar cell design and swarming models, and we consider several spectral functions with interesting interpretations such as spectral radius, algebraic connectivity, effective resistance, and condition number. We prove that the regular simplex extremizes several spectral invariants on the sphere. We also consider pointset configurations on flat tori via (i) the analogous problem on lattices and (ii) through a variety of computational experiments. For many of the objectives considered (but not all), the triangular lattice is extremal.