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. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Emery $\infty$-curvature, which motivates a generalconjecture about Bakry-Emery $\infty$-curvature.

中文翻译：

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关于 Ollivier Ricci 曲率的图的 Bonnet-Myers 不等式的刚性

我们在奥利维尔里奇曲率意义上介绍了 Bonnet-Myers 和 Lichnerowicz 锐度的概念。我们的主要结果是对所有以自我为中心的 Bonnet-Myers 锐图（超立方体、鸡尾酒会图、偶维半立方体、约翰逊图 $J(2n,n)$、Gosset 图和合适的笛卡尔积）进行分类。我们还对这个结果进行了纯粹的组合重构。我们证明 Bonnet-Myers 锐度意味着 Lichnerowicz 锐度。我们还将 Bonnet-Myers 锐度与 Bakry-Emery $\infty$-curvature 的上限联系起来，这激发了关于 Bakry-Emery $\infty$-curvature 的一般猜想。