On the notions of lower Ricci curvature bound for discrete graphs

The Ricci curvature plays an important role in Riemannian geometry. The assumption that the manifold has nonnegative Ricci curvature implies some geometric and topological constraints (For instance, the diameter of the manifold is bounded and so the manifold is compact. This is the famous Bonnet–Myers Theorem). In these notes, we present several approaches to extend this kind of results in the setting of discrete graphs, in particular Cayley graphs of finitely generated groups.