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

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

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
• Public URL: https://durham-repository.worktribe.com/output/1272160