Clique polynomials have a unique root of smallest modulus

Abstract Given an undirected graph G , let P G (z) be the polynomial P G (z)= ∑ 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.