Spatio–Spectral Limiting on Replacements of Tori by Cubes

Jeffrey A. Hogan and Joseph D. Lakey ()
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Jeffrey A. Hogan: School of Information and Physical Sciences (Mathematics), Newcastle University, Callaghan, NSW 2308, Australia
Joseph D. Lakey: Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, USA

Mathematics, 2023, vol. 11, issue 23, 1-14

Abstract: A class of graphs is defined in which each vertex of a discrete torus is replaced by a Boolean hypercube in such a way that vertices in a fixed subset of each replacement cube are adjacent to corresponding vertices of a neighboring replacement cube. Bases of eigenvectors of the Laplacians of the resulting graphs are described in a manner suitable for quantifying the concentration of a low-spectrum vertex function on a single vertex replacement. Functions that optimize this concentration on these graphs can be regarded as analogues of Slepian prolate functions that optimize concentration of a bandlimited signal on an interval in the classical setting of the real line. Comparison to the case of a simple discrete cycle shows that replacement allows for higher concentration.

Keywords: Boolean cube; discrete torus; replacement graph; spatio–spectral limiting; spectral graph theory (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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