A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian

Francesco Tudisco, Matthias Hein

Research output: Contribution to journalArticle

Abstract

We consider the nonlinear graph p-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational prin- ciple. We prove a nodal domain theorem for the graph p-Laplacian for any p 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes for p = 1. We show that the bounds are tight for p 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p-Laplacian for p > 1. If the eigenfunction associated to the k-th variational eigenvalue of the graph p-Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p ! 1.
LanguageEnglish
JournalJournal of Spectral Theory
Publication statusAccepted/In press - 18 Mar 2017

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Nodal Domain
P-Laplacian
theorems
Higher Order
eigenvectors
Graph in graph theory
Theorem
Eigenfunctions
eigenvalues
Graph Laplacian
operators
Eigenvalue
Operator

Keywords

  • nodal domain theorem
  • Cheeger inequality
  • spectral theory
  • eigenvalues

Cite this

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abstract = "We consider the nonlinear graph p-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational prin- ciple. We prove a nodal domain theorem for the graph p-Laplacian for any p 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes for p = 1. We show that the bounds are tight for p 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p-Laplacian for p > 1. If the eigenfunction associated to the k-th variational eigenvalue of the graph p-Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p ! 1.",
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A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian. / Tudisco, Francesco; Hein, Matthias.

In: Journal of Spectral Theory, 18.03.2017.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian

AU - Tudisco, Francesco

AU - Hein, Matthias

PY - 2017/3/18

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N2 - We consider the nonlinear graph p-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational prin- ciple. We prove a nodal domain theorem for the graph p-Laplacian for any p 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes for p = 1. We show that the bounds are tight for p 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p-Laplacian for p > 1. If the eigenfunction associated to the k-th variational eigenvalue of the graph p-Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p ! 1.

AB - We consider the nonlinear graph p-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational prin- ciple. We prove a nodal domain theorem for the graph p-Laplacian for any p 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes for p = 1. We show that the bounds are tight for p 1 as the bounds are attained by the eigenfunctions of the graph p-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph p-Laplacian for p > 1. If the eigenfunction associated to the k-th variational eigenvalue of the graph p-Laplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p ! 1.

KW - nodal domain theorem

KW - Cheeger inequality

KW - spectral theory

KW - eigenvalues

UR - https://arxiv.org/abs/1602.05567

M3 - Article

JO - Journal of Spectral Theory

T2 - Journal of Spectral Theory

JF - Journal of Spectral Theory

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