The independence polynomial of a simple graph \(G\) is \(\sum_A x^{| A| }\), summed over independent vertex sets \(A\subseteq V(G)\). \textit{O. J. Heilman} and \textit{E. H. Lieb} [Commun. Math. Phys. 25, 190-232 (1972; Zbl 0228.05131)], erratum [Commun. Math. Phys. 27, 166 (1972; Zbl 0238.05114)] proved that all the roots of the independence polynomial of a line graph are real. (This property does not hold for all graphs, namely a claw, i.e. \(K_{1,3}\), is a counterexample.) Y. O. Hamidoune and R. P. Stanley conjectured that the independence polynomials of clawfree graphs (i.e. those without an induced \(K_{1,3}\)) have only real roots. This is an extension of the cited result of O. J. Heilman and E. H. Lieb, as line graphs are clawfree. The paper under review proves this conjecture. A key ingredient of the proof is a new lemma about polynomials with real coefficients. A sequence of polynomials \(f_1,f_2,\dots,f_k\) are said to be compatible, if for all \(c_1,c_2,\dots,c_k\) nonnegative real numbers, \(\sum_{i=1}^k c_if_i\) has only real roots. The lemma asserts that if \(f_1,f_2,\dots,f_k\) are pairwise compatible polynomials, all with positive leading coefficients, then they are compatible.