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

D. Cushing Department of Mathematical Sciences, Durham University, United Kingdom of Great Britain and Northern Ireland S. Kamtue Department of Mathematical Sciences, Durham University, United Kingdom of Great Britain and Northern Ireland J. Koolen School of Mathematical Sciences, University of Science and Technology of China, and Wu Wen-Tsun Key Laboratory of Mathematics of CAS, Hefei, China S. Liu School of Mathematical Sciences, University of Science and Technology of China, and Wu Wen-Tsun Key Laboratory of Mathematics of CAS, Hefei, China F. Münch nstitute of Mathematics, Universität Potsdam, Germany N. Peyerimhoff Department of Mathematical Sciences, Durham University, United Kingdom of Great Britain and Northern Ireland

Combinatorics Differential Geometry mathscidoc:2203.06004

Advances in Mathematics, 369, 107188, 2020.8
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 , the Gosset graph J(2n, n) 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 θ_1=b_1-1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature.
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@inproceedings{d.2020rigidity,
  title={Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature},
  author={D. Cushing, S. Kamtue, J. Koolen, S. Liu, F. Münch, and N. Peyerimhoff},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318132513039785010},
  booktitle={Advances in Mathematics},
  volume={369},
  pages={107188},
  year={2020},
}
D. Cushing, S. Kamtue, J. Koolen, S. Liu, F. Münch, and N. Peyerimhoff. Rigidity of the Bonnet-Myers inequality for graphs with respect to Ollivier Ricci curvature. 2020. Vol. 369. In Advances in Mathematics. pp.107188. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220318132513039785010.
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