  In this paper, several nonlinear elliptic systems are investigated on graphs.
One type of the sobolev embedding theorem and a new version of the strong
maximum principle are established. Then, by using the variational method, the
existence of different types of solutions to some elliptic systems is
confirmed. Such problems extend the existence results on closed Riemann surface
to graphs and extend the existence results for one single equation on graphs
[A. Grigor'yan, Y. Lin, Y. Yang, J. Differential Equations, 2016] to nonlinear
elliptic systems on graphs. Such problems can also be viewed as one type of
discrete version of the elliptic systems on Euclidean space and Riemannian
manifold.
