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\to 1 .