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

摘要

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 theta(1) = b(1) - 1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Emery infinity-curvature, which motivates a general conjecture about Bakry-Emery infinity-curvature. (C) 2020 Elsevier Inc. All rights reserved.