  Ricci curvature was proposed by Ollivier in a general framework of metric
measure spaces, and it has been studied extensively in the context of graphs in
recent years. In this paper we prove upper bounds for Ollivier's Ricci
curvature for bipartite graphs and for the graphs with girth at least 5. We
also prove a general lower bound on the Ricci curvature in terms of the size of
the maximum matching in an appropriate subgraph. As a consequence, we
characterize the Ricci-flat graphs of girth 5. Moreover, using our general
lower bound and the Birkhoff-von Neumann theorem, we give a necessary and
sufficient condition for the structure of Ricci-flat regular graphs of girth 4.
Finally, we obtain the asymptotic Ricci curvature of random bipartite graphs
$G(n,n, p)$ and random graphs $G(n, p)$, in various regimes of $p$.
