The Laplacian on rapidly branching trees
type: Item
instance of: scholarly article
title: The Laplacian on rapidly branching trees
zbMATH Open document ID: 0856.58044
DOI: 10.1215/S0012-7094-96-08308-8
author: Koji Fujiwara
published in: Duke Mathematical Journal
publication date: 1997-02-24T00:00:00Z
review text: Let \(K(r):= \sup\{K(x, \pi): d(p, x)\geq r\}\), where \(K(x, \pi)\) is the sectional curvature of a 2-plane in \(T_x M\) and \(p\) is the basepoint of a complete simply connected negatively curved Riemannian manifold \(M\). Let \(\text{Ess}(\Delta_M)\) be the essential spectrum of the Laplacian on M. Donnelly and Li proved: Theorem. If \(\lim_{r\to \infty} K(r)= -\infty\), then \(\text{Ess}(\Delta_M)\) is empty. The author establishes an analogous theorem for graphs. Let \(\alpha_\infty\) be the isoperimetric constant at infinity of a locally finite infinite graph \(G\) and let \(\Delta_G\) be the combinatorial Laplacian. Theorem. a) \(\alpha_\infty= 1\Leftrightarrow \text{Ess}(\Delta_G)= \{1\}\). b) If \(T\) is a rapidly branching tree, then \(\text{Ess}(\Delta_T)= \{1\}\). c) Let \(T\) be an infinite tree with \(m\geq 2\). If 1 is an eigenvalue of \(\Delta_T\), then it has infinite multiplicity. The author also gives necessary and sufficient conditions in terms of the branching numbers that 1 is an eigenvalue of \(\Delta_T\) if \(T\) is an infinite symmetric tree.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 912125
zbMATH Keywords: infinite graph
zbMATH Keywords: combinatorial Laplacian
zbMATH Keywords: rapidly branching tree
zbMATH Keywords: eigenvalue
zbMATH Keywords: infinite symmetric tree
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08308-8
OpenAlex ID: W2092517882
description: scientific article; zbMATH DE number 912125