Graphical Designs and Gale Duality

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. We also prove that unless NP = coNP, there cannot be an efficient description of all minimal designs that average a fixed number of eigenspaces in a graph.

Random Walks, Equidistribution and Graphical Designs

“…Remark 6. While the inequality # {i : w i = 0} ≤ is needed in general (see [2] for examples where this bound is tight), there are examples where # {i : w i = 0} is much smaller than while still satisfying φ j , w = 0 for all 2 ≤ j ≤ .…”

“…Graphical designs were further explored by Babecki [1] and Golubev [11]. The first general Existence Theorem was recently proven by Babecki & Thomas [2] who established a bijection between positively weighted graphical designs and the faces of a generalized eigenpolytope of the graph.…”