Graphical Designs and Extremal Combinatorics

TL;DR

This paper explores the relationship between extremal independent sets, isoperimetric sets, and graphical designs in graphs, providing new examples and analyzing their properties, especially in hypercube graphs.

Contribution

It establishes that extremal independent and isoperimetric sets are extremal graphical designs and investigates their behavior under graph products and in hypercube graphs.

Findings

Extremal independent sets are extremal graphical designs.

Sets realizing the isoperimetric constant are extremal graphical designs.

Identifies a family of extremal graphical designs for hypercube graphs.

Abstract

A graphical design is a proper subset of vertices of a graph on which many eigenfunctions of the Laplacian operator have mean value zero. In this paper, we show that extremal independent sets make extremal graphical designs, that is, a design on which the maximum possible number of eigenfunctions have mean value zero. We then provide examples of such graphs and sets, which arise naturally in extremal combinatorics. We also show that sets which realize the isoperimetric constant of a graph make extremal graphical designs, and provide examples for them as well. We investigate the behavior of graphical designs under the operation of weak graph product. In addition, we present a family of extremal graphical designs for the hypercube graph.