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

  • Francesco Tudisco

    University of Strathclyde, Glasgow, UK
  • Matthias Hein

    Universität Tübingen, Germany
A nodal domain theorem and a higher-order Cheeger inequality for the graph $p$-Laplacian cover
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Abstract

We consider the nonlinear graph pp-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph pp-Laplacian for any p1p\geq 1. While for p>1p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p=2p = 2), the behavior changes for p=1p = 1. We show that the bounds are tight for p1p\geq 1 as the bounds are attained by the eigenfunctions of the graph pp-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph pp-Laplacian for p>1p > 1. If the eigenfunction associated to the kk-th variational eigenvalue of the graph pp-Laplacian has exactly kk strong nodal domains, then the higher order Cheeger inequality becomes tight as p1p\to 1.

Cite this article

Francesco Tudisco, Matthias Hein, A nodal domain theorem and a higher-order Cheeger inequality for the graph pp-Laplacian. J. Spectr. Theory 8 (2018), no. 3, pp. 883–908

DOI 10.4171/JST/216