The Peterson recurrence formula for the chromatic discriminant of a graph

TL;DR

This paper introduces a new recurrence formula for the chromatic discriminant of a graph, derived from Kac-Moody Lie algebra theory, with combinatorial proofs using acyclic orientations and spanning trees.

Contribution

It presents a novel recurrence formula for the chromatic discriminant based on Lie algebra theory, expanding the algebraic and combinatorial understanding of this graph invariant.

Findings

New recurrence formula for lpha(G) derived from Kac-Moody Lie algebras

Two bijective proofs using acyclic orientations and spanning trees

Enhanced understanding of algebraic and combinatorial interpretations of lpha(G)

Abstract

The absolute value of the coefficient of $q$ in the chromatic polynomial of a graph $G$ is known as the chromatic discriminant of $G$ and is denoted $\alpha(G)$. There is a well known recurrence formula for $\alpha(G)$ that comes from the deletion-contraction rule for the chromatic polynomial. In this paper we prove another recurrence formula for $\alpha(G)$ that comes from the theory of Kac-Moody Lie algebras. We start with a brief survey on many interesting algebraic and combinatorial interpretations of $\alpha(G)$. We use two of these interpretations (in terms of acyclic orientations and spanning trees) to give two bijective proofs for our recurrence formula of $\alpha(G)$.