  We consider a kind of nonlinear systems on a locally finite graphs $G=(V,E)$.
We prove via the mountain pass theorem that this kind of systems has a
nontrivial ground state solution which depends on the parameter $\lambda$ with
some suitable assumptions on the potentials. Moreover, we pay attention to the
concentration behavior of these solutions and prove that, as $\lambda \to
\infty$, these solutions converge to a ground state solution of a corresponding
Dirichlet problem. Finally, we also provide some numerical experiments to
illustrate our results.
