### Abstract

Language | English |
---|---|

Journal | Journal of Spectral Theory |

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

### Fingerprint

### Keywords

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

### Cite this

*Journal of Spectral Theory*.

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*Journal of Spectral Theory*.

**A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian.** / Tudisco, Francesco; Hein, Matthias.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A nodal domain theorem and a higher-order Cheeger inequality for the graph p-Laplacian

AU - Tudisco, Francesco

AU - Hein, Matthias

PY - 2017/3/18

Y1 - 2017/3/18

N2 - 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.

AB - 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.

KW - nodal domain theorem

KW - Cheeger inequality

KW - spectral theory

KW - eigenvalues

UR - https://arxiv.org/abs/1602.05567

M3 - Article

JO - Journal of Spectral Theory

T2 - Journal of Spectral Theory

JF - Journal of Spectral Theory

SN - 1664-039X

ER -