General Cheeger inequalities for p-Laplacians on graphs

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 geometr...

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Bibliographic Details
Published in:Nonlinear analysis 2016-12, Vol.147, p.80-95
Main Authors: Keller, Matthias, Mugnolo, Delio
Format: Article
Language:English
Subjects:
Cheeger inequalities
Geometry
Graphs
Inequalities
Inequality
Intrinsic metrics for Dirichlet forms
Nonlinear equations
Spectral theory of graphs
Upper bounds
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Summary: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.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2016.07.011