  By using the well-known mountain pass theorem and Ekeland's variational
principle, we prove that there exist at least two fully-non-trivial solutions
for a $(p,q)$-Kirchhoff elliptic system with the Dirichlet boundary conditions
and perturbation terms on a locally weighted and connected finite graph
$G=(V,E)$.We also present a necessary condition of the existence of
semi-trivial solutions for the system. Moreover, by using Ekeland's variational
principle and Clark's Theorem, respectively, we prove that the system has at
least one or multiple semi-trivial solutions when the perturbation terms
satisfy different assumptions. Finally, we present a nonexistence result of
solutions.
