Existence of Solutions to Mean Field Equations on Graphs

In this paper, we prove two existence results of solutions to mean field equations $Δu+e^u=ρδ_0$ and $Δu=λe^u(e^u-1)+4 π\sum_{j=1}^{M}{δ_{p_j}}$ on an arbitrary connected finite graph, where $ρ>0$ and $λ>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph.

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