We are considered with the following nonlinear Schrödinger equation −∆ u+( λa( x)+1) u= f( u) ,x∈ V, on a locally finite graph G=( V,E), where V denotes the vertex set, E denotes the edge set, λ>1 is a parameter, f( s) is asymptotically linear with respect to s at infinity, and the potential a: V→[0 ,+∞) has a nonempty well Ω. By using variational methods we prove that the above problem has a ground state solution u λ for any λ>1. Moreover, we show that as λ→∞, the ground state solution u λ converges to a ground state solution of a Dirichlet problem defined on the potential well Ω.