Growth and the spectrum of the Laplacian of an infinite graph
type: Item
instance of: scholarly article
title: Growth and the spectrum of the Laplacian of an infinite graph
zbMATH Open document ID: 0857.05070
author: Koji Fujiwara
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-11-25T00:00:00Z
review text: Let \(G\) be a locally finite, infinite graph and let \(\Delta\) be the combinatorial Laplacian. Let \(m\) and \(M\) be the infimum and the supremum of the essential spectrum of \(\Delta\); \(0\leq m\leq1\leq M\leq 2\). Let \(\mu(G)\) be the exponential growth constant of \(G\). The author shows that \[ m\leq{1-2e^{\mu/2}\over 1+e^\mu}\quad\text{and}\quad m\leq\Biggl({1-e^{\mu/2}\over 1+e^{\mu/2}}\Biggr)^2M. \] The first upper bound for \(m\) is a generalization of a result of Brooks for Riemannian manifolds. The author also derives several interesting corollaries of this result.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 927281
zbMATH Keywords: bipartite graph
zbMATH Keywords: infinite graph
zbMATH Keywords: Laplacian
zbMATH Keywords: spectrum
zbMATH Keywords: exponential growth
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225382
OpenAlex ID: W2074182921
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DOI: 10.2748/TMJ/1178225382
description: scientific article; zbMATH DE number 927281