Journal of Differential Geometry

Li-Yau inequality on graphs

Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi, and Shing-Tung Yau

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Abstract

We prove the Li-Yau gradient estimate for the heat kernel on graphs. The only assumption is a variant of the curvature-dimension inequality, which is purely local, and can be considered as a new notion of curvature for graphs. We compute this curvature for lattices and trees and conclude that it behaves more naturally than the already existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth.

We also derive Harnack inequalities and heat kernel bounds from the gradient estimate, and show how it can be used to derive a Buser-type inequality relating the spectral gap and the Cheeger constant of a graph.

Article information

Source
J. Differential Geom. Volume 99, Number 3 (2015), 359-405.

Dates
First available in Project Euclid: 25 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1424880980

Digital Object Identifier
doi:10.4310/jdg/1424880980

Mathematical Reviews number (MathSciNet)
MR3316971

Zentralblatt MATH identifier
1323.35189

Citation

Bauer, Frank; Horn, Paul; Lin, Yong; Lippner, Gabor; Mangoubi, Dan; Yau, Shing-Tung. Li-Yau inequality on graphs. J. Differential Geom. 99 (2015), no. 3, 359--405. doi:10.4310/jdg/1424880980. https://projecteuclid.org/euclid.jdg/1424880980.


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