A nodal domain theorem and a higherorder Cheeger inequality for the graph pLaplacian
Tudisco, Francesco and Hein, Matthias (2017) A nodal domain theorem and a higherorder Cheeger inequality for the graph pLaplacian. Journal of Spectral Theory. ISSN 16640403 (In Press)
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Abstract
We consider the nonlinear graph pLaplacian 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 pLaplacian 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 pLaplacian on two graphs. Finally, using the properties of the nodal domains, we prove a higherorder Cheeger inequality for the graph pLaplacian for p > 1. If the eigenfunction associated to the kth variational eigenvalue of the graph pLaplacian has exactly k strong nodal domains, then the higher order Cheeger inequality becomes tight as p ! 1.


Item type: Article ID code: 62136 Dates: DateEvent18 March 2017Published18 March 2017AcceptedKeywords: nodal domain theorem, Cheeger inequality, spectral theory, eigenvalues, Mathematics, Discrete Mathematics and Combinatorics Subjects: Science > Mathematics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 24 Oct 2017 09:37 Last modified: 12 Jul 2021 08:27 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/62136