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A gradient estimate on graphs with the \(\mathrm{CDE}(m,K)\)-condition. (English) Zbl 1372.05143

Summary: Let \(G(V,E)\) be an infinite (locally finite) graph. In this paper we prove that if \(G\) satisfies the \(\mathrm{CDE}(m,-K)\)-condition for some \(m>0\), \(K\geq 0\), and \(u:V\rightarrow \mathbb {R}\) is a positive solution to the following equation \[ \bigtriangleup_\mu u=-\lambda u, \] then \(\lambda \leq \frac{mK}{2}\) and \(u\) satisfies a gradient estimate, which is parallel to the results on complete noncompact Riemannian manifolds established by P. Li [Geometric analysis. Cambridge: Cambridge University Press (2012)], and on smooth metric measure spaces established by L. F. Wang [Ann. Global Anal. Geom. 37, No. 4, 393–402 (2010; Zbl 1190.53034)]. As byproducts, we also get two Liouville theorems and a Harnack inequality.

MSC:

05C63 Infinite graphs
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citations:

Zbl 1190.53034
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References:

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