Bakry-Emery curvature and diameter bounds on graphs
  We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from [6] and [10]. 
Shiping Liu, Florentin M\"unch, Norbert Peyerimhoff
math.DG