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) nc nz 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.