The roots of the independence polynomial of a clawfree graph

The independence polynomial of a graph G is the polynomial ∑ A x | A | , summed over all independent subsets A ⊆ V ( G ) . We prove that if G is clawfree, then all the roots of its independence polynomial are real. This extends a theorem of Heilmann and Lieb [O.J. Heilmann, E.H. Lieb, Theory of monomer–dimer systems, Comm. Math. Phys. 25 (1972) 190–232], answering a question posed by Hamidoune [Y.O. Hamidoune, On the numbers of independent k-sets in a clawfree graph, J. Combin. Theory Ser. B 50 (1990) 241–244] and Stanley [R.P. Stanley, Graph colorings and related symmetric functions: Ideas and applications, Discrete Math. 193 (1998) 267–286].