Clique polynomials have a unique root of smallest modulus

Given an undirected graph G, let P-G(z) be the polynomial P-G(z) = Sigma(n)(-1)(n)c(n)z(n), where c(n) is the number of cliques of size n in G. We show that, for every G, the polynomial P-G(z) has only one root of smallest modulus. Clique polynomials are related to trace monoids. Indeed, 1/P-G(z) is the generating function of the sequence {t(n)}, where t(n) is the number of traces of size n in the trace monoid defined by G. Our result can be applied to derive asymptotic expressions for {t(n)} and other sequences arising from the analysis of algorithms for the recognition of trace languages.