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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 $z_1, z_2,\dots, z_k$ arearbitrarily 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 anecessary and sufficient condition for the existence and uniqueness ofsolutions to the Bogomol'nyi equation.

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