In this paper, we prove two existence results of solutions to mean field equations
on an arbitrary connected finite graph, where \(\rho >0\) and \(\lambda >0\) are constants, M is a positive integer, and \(p_1,\ldots ,p_M\) are arbitrarily chosen distinct vertices on the graph.
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Y. Lin is supported by the National Science Foundation of China (Grant No. 11271011 and 11761131002), S.-T. Yau is supported by the NSF DMS-0804. Part of the work was done when Y. Lin visited the Harvard CMSA in 2018.
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Communicated by H. T. Yau.
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Huang, A., Lin, Y. & Yau, S. Existence of Solutions to Mean Field Equations on Graphs. Commun. Math. Phys. 377, 613–621 (2020). https://doi.org/10.1007/s00220-020-03708-1