Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I ( G , x ) = ∑ k ≥ 0 i k x k , where i k is the number of independent sets of cardinality k in G . We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, even when restricted to circulants. We provide a formula for the independence polynomial of a certain family of circulants, and its complement. As an application, we derive a formula for the number of chords in an n -tet musical system (one where the ratio of frequencies in a semitone is 2 1 / n ) without ‘close’ pitch classes.