Independence polynomials of circulants with an application to music

The independence polynomial   of a graph GG is the generating function I(G,x)=∑k≥0ikxkI(G,x)=∑k≥0ikxk, where ikik is the number of independent sets of cardinality kk in GG. 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 nn-tet musical system (one where the ratio of frequencies in a semitone is 21/n21/n) without ‘close’ pitch classes.