Chromatic polynomials of graphs from Kac-Moody algebras

TL;DR

This paper links the chromatic polynomial of a graph to the q-Kostant partition function of a related Kac-Moody algebra, providing a new algebraic interpretation and computational approach.

Contribution

It introduces a novel interpretation of chromatic polynomials via Kac-Moody Lie algebras and derives a new formula using root multiplicities and the bond lattice.

Findings

Chromatic polynomial expressed as q-Kostant partition function

New realization of chromatic polynomial as weighted sum of paths

Connection between graph coloring and Lie algebra root multiplicities

Abstract

We give a new interpretation of the chromatic polynomial of a simple graph G in terms of the Kac-Moody Lie algebra with Dynkin diagram G. We show that the chromatic polynomial is essentially the q-Kostant partition function of this Lie algebra evaluated on the sum of the simple roots. Applying the Peterson recurrence formula for root multiplicities, we obtain a new realization of the chromatic polynomial as a weighted sum of paths in the bond lattice of G.