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Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature
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,∞) with respect to the unnormalized Laplacian, then |V|≤ 2d, 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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BibTeX
@misc{ding2026entropy,
title = {{Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature}},
author = {Yanlong Ding and Shiping Liu and Chiyu Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23417},
eprint = {2610.23417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23417},
}
RIS
TY - UNPB
TI - Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature
AU - Yanlong Ding
AU - Shiping Liu
AU - Chiyu Zhou
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23417
UR - https://arxiv.org/abs/2610.23417
AB - 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,∞) with respect to the unnormalized Laplacian, then |V|≤ 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.
ER -