Existence and uniqueness of solutions to Bogomol'nyi equations on graphs

Let $G=(V,E)$ be a connected finite graph. We study the Bogomol'nyi equation \begin{equation*} \Delta u= \mathrm{e}^{u}-1 +4 \pi \sum_{s=1}^{k} n_s \delta_{z_{s}} \quad \text { on } \quad G, \end{equation*} where $z_1, z_2,\dots, z_k$ are arbitrarily chosen distinct vertices on the graph, $n_j$ is a positive integer, $j=1,2,\cdots, k$ and $\delta_{z_{s}}$ is the Dirac mass at $z_s$. We obtain a necessary and sufficient condition for the existence and uniqueness of solutions to the Bogomol'nyi equation.