  We investigate the existence of ground state solutions for a
$(p,q)$-Laplacian system with $p,q>1$ and potential wells on a weighted locally
finite graph $G=(V,E)$. By making use of the method of Nehari manifold and the
Lagrange multiplier rule, we prove that if the nonlinear term $F$ takes on the
super-$(p, q)$-linear growth and the potential functions $a(x)$ and $b(x)$
satisfy some suitable conditions, then for any fixed parameter $\lambda\geq1$,
the system is provided with a ground state solution $(u_\lambda, v_\lambda)$.
Additionally, we set up the convergence property of the solutions set
$\{(u_\lambda, v_\lambda)\}$ when $\lambda \rightarrow +\infty$.
