We consider the nonlinear graph -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 -Laplacian for any . While for the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (), the behavior changes for . We show that the bounds are tight for as the bounds are attained by the eigenfunctions of the graph -Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higher-order Cheeger inequality for the graph -Laplacian for . If the eigenfunction associated to the -th variational eigenvalue of the graph -Laplacian has exactly strong nodal domains, then the higher order Cheeger inequality becomes tight as .
Cite this article
Francesco Tudisco, Matthias Hein, A nodal domain theorem and a higher-order Cheeger inequality for the graph -Laplacian. J. Spectr. Theory 8 (2018), no. 3, pp. 883–908DOI 10.4171/JST/216