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 Lₕ that are not necessarily lower semi-bounded. As a corollary of the main result, we show that Lₕ is essentially self-adjoint if the potential V satisfies V (x) -b₁-b₂ρ (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).
Ognjen Milatovic (Wed,) studied this question.