Bakry-Emery curvature on graphs and diameter-bounds

We give diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result is using only curvature and maximal vertex degree. Rigidity of this diameter bound characterizes the hypercube. 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.