Bakry Emery curvature and diameter bounds on graphs
We prove finiteness and diameter bounds for graphs having a positive Ricci curvature bound in the Bakry Emery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second 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 Fathi and Shu (Bernoulli 24(1):672 698, 2018) and Horn et al. (J fur die reine und angewandte Mathematik (Crelle's J) 2017, https:/doi.org/10.1515/crelle 2017 0038) and solve a conjecture from Cushing et al. (Bakry Emery curvature functions of graphs, 2016)