Essential self-adjointness of non-semibounded Schr\"odinger operators on infinite graphs

We work in the setting of infinite, not necessarily locally finite, weighted graphs. We give a sufficient condition for the essential self-adjointness of (discrete) Schr\"odinger operators $\mathcal{L}_{V}$ that are not necessarily lower semi-bounded. As a corollary of the main result, we show that $\mathcal{L}_{V}$ is essentially self-adjoint if the potential $V$ satisfies $V(x)\geq -b_1-b_2[\rho(0,x)]^2$, for all vertices $x$, where $o$ is a fixed vertex, $b_1$ and $b_2$ are non-negative constants, and $\rho$ is an intrinsic metric of finite jump size, such that the restriction of the weighted vertex degree to every ball corresponding to $\rho$ is bounded (not necessarily uniformly bounded).