Holomorphic quadratic differentials on graphs and the chromatic polynomial

TL;DR

This paper explores the connection between holomorphic quadratic differentials on graphs, reactive power in circuits, and the chromatic polynomial, providing new methods to evaluate the polynomial at negative integers.

Contribution

It establishes a novel link between holomorphic quadratic differentials and the chromatic polynomial, offering an explicit integral expression for its values at negative integers.

Findings

Chromatic polynomial at negative integers equals the degree of a rational map.

Explicit integral expression for the chromatic polynomial at negative integers.

Connection between holomorphic differentials and graph coloring properties.

Abstract

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 $\chi(-k)$.