Open Access
March 2015 Li-Yau inequality on graphs
Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi, Shing-Tung Yau
J. Differential Geom. 99(3): 359-405 (March 2015). DOI: 10.4310/jdg/1424880980


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.


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Frank Bauer. Paul Horn. Yong Lin. Gabor Lippner. Dan Mangoubi. Shing-Tung Yau. "Li-Yau inequality on graphs." J. Differential Geom. 99 (3) 359 - 405, March 2015.


Published: March 2015
First available in Project Euclid: 25 February 2015

zbMATH: 1323.35189
MathSciNet: MR3316971
Digital Object Identifier: 10.4310/jdg/1424880980

Rights: Copyright © 2015 Lehigh University

Vol.99 • No. 3 • March 2015
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