### 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 language | English |
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Journal | Journal of Spectral Theory |

Publication status | Accepted/In press - 18 Mar 2017 |

### Keywords

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

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## Cite this

Tudisco, F., & Hein, M. (Accepted/In press). A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian.

*Journal of Spectral Theory*.