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

Francesco Tudisco, Matthias Hein

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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.
Original languageEnglish
JournalJournal of Spectral Theory
Publication statusAccepted/In press - 18 Mar 2017

Keywords

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

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