The Poisson's Problems on graphs

In this paper we study the problem $$\{\begin{array}{ll}-{\mathrm{\Delta }}_{d}u={\mu }_0& \text{ in }G\\ u=0& \text{ on }\mathrm{\partial }G\end{array}$$ where, ${\mathrm{\Delta }}_d$ represent the discret Laplacian, and ${\mu }_0$ it is a measure defined in the vertex of the graph $G=(V,E)$. Here $V$ defined the vertex of the graph, $E$ its edges and $\mathrm{\partial }G$ its boundary. We prove that this problem has an unique solution by using an adaption of the Perron’s method for the graphs by using an idea known as Balayage.