Generalized Designs on Graphs: Sampling, Spectra, Symmetries

TL;DR

This paper extends the concept of spherical designs to finite graphs, introducing 'graphical designs' that are evenly distributed vertex subsets capturing graph symmetries, with results on their structure and examples.

Contribution

It introduces a spectral definition of graphical designs, linking them to graph symmetries and distribution, and provides structural results and methods to find such designs.

Findings

Good graphical designs are either large or have neighborhoods with exponential growth.

Examples of graphical designs are provided.

Methods for identifying graphical designs are discussed.

Abstract

Spherical Designs are finite sets of points on the sphere $\mathbb{S}^{d}$ with the property that the average of certain (low-degree) polynomials in these points coincides with the global average of the polynomial on $\mathbb{S}^{d}$. They are evenly distributed and often exhibit a great degree of regularity and symmetry. We point out that a spectral definition of spherical designs easily transfers to finite graphs -- these 'graphical designs' are subsets of vertices that are evenly spaced and capture the symmetries of the underlying graph (should they exist). Our main result states that good graphical designs either consist of many vertices or their neighborhoods have exponential volume growth. We show several examples, describe ways to find them and discuss problems.