In this article, we study the well-posedness (uniqueness and existence of solutions) of nonlinear elliptic Partial Differential Equations (PDEs) on a finite graph. These results are obtained using the discrete comparison principle and connectivity properties of the graph. This work is in the spirit of the theory of viscosity solutions for partial differential equations. The equations include the graph Laplacian, the $p$-Laplacian, the Infinity Laplacian, and the Eikonal operator on the graph.
"Nonlinear elliptic Partial Differential Equations and p-harmonic functions on graphs." Differential Integral Equations 28 (1/2) 79 - 102, January/February 2015.