General Cheeger inequalities for p-Laplacians on graphs. (December 2016)

• Record Type: Journal Article
• Authors: Keller, Matthias; Mugnolo, Delio
• Is Part Of: Nonlinear analysis. Volume 147(2016)
• Page Start: 80
• Page End: 95
• Publication Date: 2016-12
• Subjects: Cheeger inequalities -- Spectral theory of graphs -- Intrinsic metrics for Dirichlet forms; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.07.011
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We prove Cheeger inequalities for p -Laplacians on finite and infinite weighted graphs. Unlike in previous works, we do not impose boundedness of the vertex degree, nor do we restrict ourselves to the normalized Laplacian and, more generally, we do not impose any boundedness assumption on the geometry. This is achieved by a novel definition of the measure of the boundary which uses the idea of intrinsic metrics. For the non-normalized case, our bounds on the spectral gap of p -Laplacians are already significantly better for finite graphs and for infinite graphs they yield non-trivial bounds even in the case of unbounded vertex degree. We, furthermore, give upper bounds by the Cheeger constant and by the exponential volume growth of distance balls.