  The Glazman-Povzner-Wienholtz theorem states that the completeness of a
manifold, when combined with the semiboundedness of the Schr\"odinger operator
$-\Delta + q$ and suitable local regularity assumptions on $q$, guarantees its
essential self-adjointness. Our aim is to extend this result to Schr\"odinger
operators on graphs. We first obtain the corresponding theorem for
Schr\"odinger operators on metric graphs, allowing in particular distributional
potentials $q\in H^{-1}_{\rm loc}$. Moreover, we exploit recently discovered
connections between Schr\"odinger operators on metric graphs and weighted
graphs in order to prove a discrete version of the Glazman-Povzner-Wienholtz
theorem.
