On two properties of positively perturbed discrete Schr\"odinger operators

We show that if we start from a symmetric lower semi-bounded Schr\"odinger operator $\mathcal{H}$ on finitely supported functions on a discrete weighted graph (satisfying certain conditions), apply the Friedrichs construction to get a self-adjoint extension $H$, and then perturb $H$ by a non-negative function $W$, then the resulting form-sum $H\widetilde{+}W$ coincides with the Friedrichs extension of $\mathcal{H}+W$. Additionally, we consider a non-negative perturbation of an essentially self-adjoint lower semi-bounded Schr\"odinger operator $H$ on a discrete weighted graph. We show that, under certain conditions on the graph and the perturbation, the essential self-adjointness of $H$ remains stable under the given perturbation.