  We consider the porous medium equation (PME) on a locally finite graph and
identify suitable curvature-dimension (CD) conditions under which a discrete
version of the fundamental Aronson-B\'enilan estimate holds true for positive
solutions of the PME. We also show that these estimates allow to prove Harnack
inequalities which are structurally similar to the continuous case. The new CD
conditions are illustrated with several concrete examples, e.g.\ complete and
chain-like graphs.
