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

Abstract

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.

Characterizations of Forman curvature

“…A fundamental consequence of uniformly positive Ricci curvature is the Bonnet Myers diameter bound. On graphs, this is well known for Ollivier curvature [20] and Bakry Emery curvature [12], as well as their rigidity results [3,13]. A diameter bound in terms of entropic curvature is also known [5,Proposition 3.4], but looks somewhat different.…”

Sparse expanders have negative curvature

“…Consequences in terms of geometry, mixing, and concentration of measure have been massively investigated, and quantified by a variety of functional inequalities. The literature is too vast for an exhaustive account, and we refer the reader to the seminal papers [42,43,34,30], the survey [44], and the more recent works [24,41,21,32,40] for details, variations, references, and open problems. In particular, the present work was motivated by the following long-standing question, due to Naor and Milman, and publicized by Ollivier [44,Problem T].…”

Cheng maximal diameter theorem for hypergraphs

“…Moreover, under the same assumption and if Diam M = π/ √ K, then M is isometric to the n dimensional sphere S n (K −1/2 ), which is known as Cheng's maximal diameter theorem. These theorem have been extended to many other situations(see for instance [7-10, 15, 20-23, 28, 30, 31, 36] for Bonnet-Myers theorem and [12,17,19,29,32,33] for rigidity theorem). Here we focus on Bonnet-Myers type and Cheng's maximal diameter theorems for CD spaces and RCD spaces.…”

“…And sometimes Cheng's maximal diameter theorem is also given. For instance, the maximal diameter theorem on graphs and directed graphs under positive Ricci curvature in the sense of Lin-Lu-Yau are proven in [12,32]. Note that Laplacian related such notions are linear though sometimes those operator is not symmetric and the corresponding Dirichlet form is nonlocal.…”