A construction of spherical designs from finite graphs with the theory of crystal lattice

TL;DR

This paper proposes a method to construct spherical designs using finite graphs and crystal lattice theory, involving the realization of crystal lattices as maximal Abelian coverings and analyzing their vector sets.

Contribution

It introduces a novel construction of spherical designs from finite graphs via crystal lattices and explores their properties through numerical experiments and conjectures.

Findings

Successful construction of spherical designs from various finite graphs

Identification of conditions for vectors to form spherical designs

Numerical evidence supporting the proposed construction method

Abstract

We want to introduce a construction of spherical designs from finite graphs with the theory of crystal lattice. We start from a finite graph, and we consider standard realization of the crystal lattices as the maximal Abelian covering of the graph. Then, we take the set of vectors which form the crystal lattice. If every vector has the same norm, then we can consider a finite set on Euclidean sphere, and then we get a spherical design. In this paper, we observe the results by numerical calculations. We tried constructing vectors from various finite graphs, strongly regular graphs, distance regular graphs, and so on. We also introduce some facts and conjectures.