Existence of ground state solutions to some Nonlinear Schrödinger equations on lattice graphs

In this paper, we study the nonlinear Schrödinger equation $$ -\Delta u+V(x)u=f(x,u) on lattice graph $$\mathbb {Z}^{N}$$ . Using Nehari method, prove that when f satisfies some growth conditions and potential function V is periodic or bounded, above admits a ground state solution. Moreover, extend our results from to quasi-transitive graphs.