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

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 $p→1$.

Please cite this work as:

@article{tudisco2016nodal,
  title={A nodal domain theorem and a higher-order {C}heeger inequality for the graph $p$-{L}aplacian},
  author={Tudisco, Francesco and Hein, Matthias},
  journal={EMS Journal of Spectral Theory},
  volume={8},
  issue={3},
  pages={883--908},
  year={2018},
  publisher={European Mathematical Society}
}

Keywords: nonlinear eigenvalues Graph p-Laplacian Graph Laplacian eigenvalues nodal domains variational eigenvalues Cheeger inequality spectral clustering networks