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

### Francesco Tudisco

University of Strathclyde, Glasgow, UK### Matthias Hein

Universität Tübingen, Germany

## Abstract

We consider the nonlinear graph $p$-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 $p$-Laplacian for any $p\geq 1$. 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 $p\geq 1$ 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 $p\to 1$.

## Cite this article

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

DOI 10.4171/JST/216