# Existence of Solutions to Mean Field Equations on Graphs

## TL;DR

This paper establishes the existence of solutions to specific mean field equations on finite graphs, expanding the understanding of nonlinear PDEs in discrete structures with potential applications in network theory.

## Contribution

It provides the first existence results for mean field equations with delta sources on arbitrary finite graphs, generalizing previous continuous domain results.

## Key findings

- Proved existence of solutions for the first mean field equation with a delta source.
- Established solutions for the second mean field equation with multiple delta sources.
- Extended the theory of nonlinear equations to discrete graph settings.

## Abstract

In this paper, we prove two existence results of solutions to mean field equations $$\Delta u+e^u=\rho\delta_0$$ and $$\Delta u=\lambda e^u(e^u-1)+4 \pi \sum_{j=1}^{M}{\delta_{p_j}}$$ on an arbitrary connected finite graph, where $\rho>0$ and $\lambda>0$ are constants, $M$ is a positive integer, and $p_1,...,p_M$ are arbitrarily chosen vertices on the graph.