  We construct optimal Hardy weights to subcritical energy functionals $h$
associated with quasilinear Schr\"odinger operators on locally finite graphs.
Here, optimality means that the weight $w$ is the largest possible with respect
to a partial ordering, and that the corresponding shifted energy functional
$h-w$ is null-critical. Moreover, we show a decay condition of Hardy weights in
terms of their integrability with respect to certain integral weights. As an
application of the decay condition, we show that null-criticality implies
optimality near infinity. We also briefly discuss an uncertainty-type principle
and a Rellich-type inequality.
