The Peterson recurrence formula for the chromatic discriminant of a graph

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 $\alpha(G)$.