We study Hardy inequalities for p-Schrödinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for p-Schrödinger operators via a generalized capacity. The novel ingredient in the proof is the demonstration that the simplified energy of the p-Schrödinger energy functional is compatible with certain normal contractions. As a consequence, we obtain a necessary integrability criterion for Hardy weights. Finally, using some tools of criticality theory, we investigate the existence of minimizers in the Hardy inequalities and discuss relations to Cheeger type estimates.
