  We introduce a new version of a curvature-dimension inequality for
non-negative curvature. We use this inequality to prove a logarithmic Li-Yau
inequality on finite graphs. To formulate this inequality, we introduce a
non-linear variant of the calculus of Bakry and \'Emery. In the case of
manifolds, the new calculus and the new curvature-dimension inequality coincide
with the common ones. In the case of graphs, they coincide in a limit. In this
sense, the new curvature-dimension inequality gives a more general concept of
curvature on graphs and on manifolds. We show that Ricci-flat graphs have a
non-negative curvature in this sense. Moreover, a variety of non-logarithmic
Li-Yau type gradient estimates can be obtained by using the new Bakry-\'Emery
type calculus. Furthermore, we use these Li-Yau inequalities to derive Harnack
inequalities on graphs.
