Extremal properties of eigenvalues for a metric graph.
type: Item
instance of: scholarly article
title: Extremal properties of eigenvalues for a metric graph.
zbMATH Open document ID: 1074.34078
author: Leonid Friedlander
publication date: 2005-04-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116184
review text: This paper deals with optimal lower bounds for the eigenvalues of the Laplace operator on a metric graph. Let \(\Gamma=(V,E)\) be a connected finite graph, where \(V\) is the set of its vertices, and \(E\) is the set of its edges. Each edge \(e\) is considered as a segment of certain length \(l(e)>0\) and call \(G=(\Gamma, (l(e)))\) a metric graph. The length of the metric graph \(G\) is defined as the sum of the length of all edges of \(G\). Let \(H_{k}\) stand for the star with \(k\) edges with equal length. For \(j\in{\mathbb{N}}\), let \(\lambda_{j}(G)\) be the \(j\)th eigenvalue of the Laplace operator on \(G\) subject to the Kirchhoff conditions on the vertices of \(G\). It is assumed that there are no vertices of degree 2 in \(G\). The author proves that \[ \mu_{j}(G)\geq\frac{\pi^{2}j^{2}}{4l(G)}\quad\text{for}\quad j\geq 2 \] and the equality holds if and only if \(G\) is a segment when \(j=2\) and \(G=H_{j}\) when \(j\geq 3\).
reviewed by: Kazushi Yoshitomi
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 2162470
zbMATH Keywords: eigenvalues
zbMATH Keywords: metric graph
zbMATH Keywords: symmetrization
MaRDI profile type: Publication
OpenAlex ID: W2533535690
cites work: Spectral geometry: direct and inverse problems. With an appendix by G. Besson
cites work: Q3330067
cites work: Q3319376
cites work: Q3828630
cites work: Quantum graphs: I. Some basic structures
cites work: Isoperimetric Inequalities in Mathematical Physics. (AM-27)
DOI: 10.5802/AIF.2095
published in: Annales de l’institut Fourier
Recommended article: On extremal eigenvalues of the graph Laplacian *
Recommended article: Eigenvalues and extremal degrees of graphs
Recommended article: On the extreme eigenvalues of regular graphs.
Recommended article: On the extensional eigenvalues of graphs
Recommended article: Extremal metrics on graphs. I
Recommended article: Extrema of graph eigenvalues
Recommended article: Extremal Graph Realizations and Graph Laplacian Eigenvalues
Recommended article: Some properties of the distance Laplacian eigenvalues of a graph
Recommended article: Extreme eigenvalues of nonregular graphs
Recommended article: Eigenvalues and the diameter of graphs
description: scientific article; zbMATH DE number 2162470