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

Cushing, David and Kamtue, Supanat and Liu, Shiping and Muench, Florentin and Peyerimhoff, Norbert (2020) 'Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature.', Advances in mathematics., 360 . p. 107188.


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

Item Type:Article
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Available under License - Creative Commons Attribution Non-commercial No Derivatives.
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Publisher statement:© 2020 This manuscript version is made available under the CC-BY-NC-ND 4.0 license
Date accepted:24 April 2020
Date deposited:26 April 2020
Date of first online publication:08 May 2020
Date first made open access:08 May 2021

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