Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I(G, x) = Sigma(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. (c) 2008 Elsevier B.V. All rights reserved.