This paper is devoted to the analysis of the following nonlinear Schrödinger equation defined on a locally finite graph \(G = (V, E)\) : \(\begin{aligned} -\Delta u+h(x)u=f(u)+g(x),\,\,\, x\in V, \end{aligned}\) where g is a perturbed term. Under more general conditions, we use variational methods to prove the existence of both negative and positive energy solutions. Our main contribution is establishing the asymptotic behavior as the perturbation tends to zero: the negative energy solution converges to zero, while the positive energy solution converges to a nontrivial solution of the unperturbed problem. This provides the first such asymptotic analysis for this class of problems on graphs.
