  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 $\alpha(G)$.
