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.
Highlights
Let G be a d-regular graph on the vertex set V of size n with no loops or multiple edges
We study a certain subclass of graphical designs, which we baptize extremal, and connect it to a well-developed branch of extremal combinatorics
We show that extremal graphical designs can be found in maximal independent sets and in subsets realizing the isoperimetric constant, and provide infinite families of examples arising naturally in extremal combinatorics
Summary
Let G be a d-regular graph on the vertex set V of size n with no loops or multiple edges. As it is pointed out in the same paper, there is some ambiguity when eigenvalues have large multiplicity. We show that extremal graphical designs can be found in maximal independent sets and in subsets realizing the isoperimetric constant, and provide infinite families of examples arising naturally in extremal combinatorics. Pesenson introduces a notion of a Paley-Wiener space for a graph and proves analogs of the classical results of the Paley-Wiener theory for certain families of graphs He proves that a for every function on a subset of vertices of a graph, there exists a low-frequency function on the whole graph that coincides with the given function on the subset.
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