# 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}
}
• Published 28 February 2018
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
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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