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Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature

Cushing, David; Kamtue, Supanat; Liu, Shiping; Muench, Florentin; Peyerimhoff, Norbert

Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature Thumbnail


Authors

David Cushing

Shiping Liu

Florentin Muench



Abstract

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs J(2n, n), the Gosset graph and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness and classify all distance-regular Lichnerowicz sharp graphs under the additional condition θ1=b1−1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature

Citation

Cushing, D., Kamtue, S., Liu, S., Muench, F., & Peyerimhoff, N. (2020). Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature. Advances in Mathematics, 360, Article 107188. https://doi.org/10.1016/j.aim.2020.107188

Journal Article Type Article
Acceptance Date Apr 24, 2020
Online Publication Date May 8, 2020
Publication Date Aug 5, 2020
Deposit Date Apr 26, 2020
Publicly Available Date May 8, 2021
Journal Advances in Mathematics
Print ISSN 0001-8708
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 360
Article Number 107188
DOI https://doi.org/10.1016/j.aim.2020.107188

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