  We study superharmonic functions for Schr\"odinger operators on general
weighted graphs. Specifically, we prove two decompositions which both go under
the name Riesz decomposition in the literature. The first one decomposes a
superharmonic function into a harmonic and a potential part. The second one
decomposes a superharmonic function into a sum of superharmonic functions with
certain upper bounds given by prescribed superharmonic functions. As
application we show a Brelot type theorem.
