The spectral geometry of \(k\)-regular groups
type: Item
instance of: scholarly article
title: The spectral geometry of \(k\)-regular groups
zbMATH Open document ID: 0772.58061
DOI: 10.1007/BF03041067
author: Robert Brooks
published in: Journal d'Analyse Mathématique
publication date: 1993-06-29T00:00:00Z
review text: A \(k\)-regular graph is a graph with the property that the degree of each vertex is equal to \(k\). One way to produce such graphs is via coset spaces of finitely generated groups. In general we can associate to any graph a discrete Laplacian. This paper studies the relationship between the spectral properties of the Laplacian on a \(k\)-regular graph and \(\Gamma\) and its (combinatorial) geometry. In particular let \(N(\Gamma,m)\) denote the number of closed paths in \(\Gamma\) of length \(m\). The first result relates the first eigenvalue of the Laplacian to the asymptotic behaviour of \(N(\Gamma,m)\) as \(m\) goes to \(\infty\). This is part of a more general picture. The author considers the function \[ f_ \Gamma(x) = \sum_ m mN(\Gamma,m)x^ m. \] This function turns out to be rational and the paper contains a description of it in terms of the eigenvalues of the Laplacian.
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 221306
zbMATH Keywords: spectral geometry
zbMATH Keywords: regular \(K\)-graphs
zbMATH Keywords: Laplacian
zbMATH Keywords: eigenvalues
reviewed by: Liviu I. Nicolaescu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03041067
OpenAlex ID: W2148907119
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description: scientific article; zbMATH DE number 221306