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

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. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature.

Geometric and spectral properties of directed graphs under a lower Ricci curvature bound

“…In the undirected case, we also have a further work on diameter comparisons (see [14]). To prove our volume comparisons, we prepare the following lemma:…”

Curvatures, graph products and Ricci flatness

“…We finish this subsection with the following upper curvature bounds for κ 0 and κ LLY : Theorem 2.6 (see [13,Theorem 4] and [7,Proposition 2.7]). Let G = (V, E) be d-regular and {x, y} ∈ E. Then…”

Reflective Graphs, Ollivier curvature, effective diameter, and rigidity

“…It is well known that a positive lower curvature bound κ ≥ K implies an upper diameter bound, see [19,24,26], given by diam(G) := max x,y d(x, y) ≤ 2 max Deg K . In [4] it was investigated for which graphs equality holds. It turned out that under the additional assumption that for all x ∈ V there exists y ∈ V with d(x, y) = diam(G), the graphs for which holds equality are precisely the following: Cocktail party graphs, Johnson graphs J(2n, n), halved cubes on 2 • 4 n vertices, Gosset graph, and Cartesian products of the mentioned graphs with same curvature.…”

“…K = λ where K is the minimal curvature, and λ is the smallest positive eigenvalue of −∆. We then use the classification of distance regular Lichnerowicz sharp graphs with an additional spectral condition from [4,Theorem 6.5] to classify reflective graphs.…”