Essential self-adjointness of non-semibounded Schrödinger 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) Schrödinger operators LV that are not necessarily lower semi-bounded. As a corollary of the main result, we show that LV is essentially self-adjoint if the potential V satisfies V(x)≥ -b₁-b₂[ρ(0,x)]², for all vertices x, where o is a fixed vertex, b₁ and b₂ are non-negative constants, and $ρ$ is an intrinsic metric of finite jump size, such that the restriction of the weighted vertex degree to every ball corresponding to $ρ$ is bounded (not necessarily uniformly bounded).