Inner-Outer Curvatures, Ricci-Ollivier Curvature and Volume Growth of Graphs

Abstract

We are concerned with the study of different notions of curvature on graphs. We show that if a graph has stronger innerouter curvature growth than a model graph, then it has faster volume growth too. We also study the relationhips of volume growth with other kind of curvatures, such as the Ollivier-Ricci curvature.

Maximal diameter theorem for directed graphs of positive Ricci curvature

“…Recently, there are some attempts to introduce the notion of the Ricci curvature for discrete spaces, and the Ricci curvature for (undirected) graphs introduced by Lin-Lu-Yau [11] is one of the well-studied objects (see also a pioneering work of Ollivier [14]). It is well-known that a lower Ricci curvature bound of them leads us to various geometric and analytic consequences (see e.g., [1], [2], [3], [7], [9], [11], [13], [16], and so on). In [11], they have provided a discrete analogue of Bonnet-Myers theorem in Riemannian geometry.…”