Holomorphic quadratic differentials on graphs and the chromatic polynomial

@article{Kenyon2020HolomorphicQD,
  title={Holomorphic quadratic differentials on graphs and the chromatic polynomial},
  author={R. Kenyon and Wai Yeung Lam},
  journal={J. Comb. Theory, Ser. A},
  year={2020},
  volume={170}
}
We study "holomorphic quadratic differentials" on graphs. We relate them to the reactive power in an LC circuit, and also to the chromatic polynomial of a graph. Specifically, we show that the chromatic polynomial $\chi$ of a graph $G$, at negative integer values, can be evaluated as the degree of a certain rational mapping, arising from the defining equations for a holomorphic quadratic differential. This allows us to give an explicit integral expression for $\chi(-k)$. 

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