  We generalize two embedding theorems and investigate the existence and
multiplicity of nontrivial solutions for a $(p,q)$-Laplacian coupled system
with perturbations and two parameters $\lambda_1$ and $\lambda_2$ on locally
finite graph. By using the Ekeland's variational principle, we obtain that
system has at least one nontrivial solution when the nonlinear term satisfies
the sub-$(p,q)$ conditions. We also obtain a necessary condition for the
existence of semi-trivial solutions to the system. Moreover, by using the
mountain pass theorem and Ekeland's variational principle, we obtain that
system has at least one solution of positive energy and one solution of
negative energy when the nonlinear term satisfies the super-$(p,q)$ conditions
which is weaker than the well-known Ambrosetti-Rabinowitz condition.
Especially, in all of the results, we present the concrete ranges of the
parameters $\lambda_1$ and $\lambda_2$.
