Ground state solutions for asymptotically linear Schrödinger equations on locally finite graphs

We are considered with the following nonlinear Schrödinger equation: on a locally finite graph , where denotes the vertex set, denotes the edge set, is a parameter, is asymptotically linear with respect to at infinity, and the potential has a nonempty well . By using variational methods, we prove that the above problem has a ground state solution for any . Moreover, we show that as , the ground state solution converges to a ground state solution of a Dirichlet problem defined on the potential well .