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

Francesco Tudisco,
Matthias Hein,

**EMS Journal of Spectral Theory**,
8 :
883--908
(2018)

### 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}
}
```

**Links:**
doi
arxiv

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