Graphical Designs and Gale Duality

TL;DR

This paper explores the relationship between graphical designs in regular graphs and Gale duality, providing a geometric framework to compute and optimize these designs, with applications to specific graph families.

Contribution

It establishes a bijection between positively weighted graphical designs and faces of a generalized eigenpolytope using Gale duality, enabling new computational approaches.

Findings

Connected graphical designs to faces of eigenpolytopes.

Computed smallest designs for specific Cayley graphs.

Provided bounds for designs in hypercube graphs.

Abstract

A graphical design is a subset of graph vertices such that the weighted averages of certain graph eigenvectors over the design agree with their global averages. We use Gale duality to show that positively weighted graphical designs in regular graphs are in bijection with the faces of a generalized eigenpolytope of the graph. This connection can be used to organize, compute and optimize designs. We illustrate the power of this tool on three families of Cayley graphs -- cocktail party graphs, cycles, and graphs of hypercubes -- by computing or bounding the smallest designs that average all but the last eigenspace in frequency order.