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arXiv · 2610.23417

Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature

Yanlong Ding·Shiping Liu·Chiyu Zhou
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Abstract

We establish a volume bound for weighted graphs with positive Bakry--Émery curvature through entropy estimates for the heat flow and characterize the equality case. As a consequence, if a connected unweighted graph $G=(V,E)$ satisfies the Bakry--Émery curvature--dimension condition $CD(2,\infty)$ with respect to the unnormalized Laplacian, then $$ |V|\le 2^d, $$ where $d$ denotes the maximum degree. Equality holds if and only if $G$ is a hypercube. We also establish an analogous volume rigidity result for the normalized Laplacian.

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Yanlong Ding, Shiping Liu, Chiyu Zhou. 2026-10-16. Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature. https://arxiv.org/abs/2610.23417

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