Li-Yau inequality for unbounded Laplacian on graphs
Chao Gong, Yong Lin, Shuang Liu, Shing-Tung Yau
TL;DR
This paper establishes a Li-Yau inequality for unbounded Laplacians on complete weighted graphs under curvature-dimension conditions, leading to new results like Harnack inequalities, heat kernel bounds, and eigenvalue estimates.
Contribution
It introduces the first Li-Yau inequality for unbounded Laplacians on graphs under curvature-dimension conditions, with several key applications.
Findings
Derived Li-Yau inequality for unbounded Laplacian on graphs
Established Harnack inequality and heat kernel bounds
Provided Cheng's eigenvalue estimate for graphs
Abstract
In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality , which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, heat kernel bounds and Cheng's eigenvalue estimate. These are first kind of results on this direction for unbounded Laplacian on graphs.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds