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

Aronson-B\'{e}nilan and Harnack estimates for the discrete porous medium equation