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

TL;DR

This paper establishes a nodal domain theorem for the nonlinear graph p-Laplacian, analyzes the behavior for different p values, and proves a higher-order Cheeger inequality, extending spectral graph theory to nonlinear operators.

Contribution

It introduces a nodal domain theorem for the graph p-Laplacian for all p ≥ 1 and derives a higher-order Cheeger inequality, generalizing classical results to nonlinear settings.

Findings

Bounds on nodal domains are tight for p ≥ 1.

Behavior of nodal domains differs at p=1 from p>1.

Higher-order Cheeger inequality becomes tight as p approaches 1.

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 $p\rightarrow 1$.