Roots of independence polynomials of well covered graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let i(k) denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = Sigma i(k)x(k). In particular, we show that if G is a well covered graph with independence number beta, then all the roots of i(G, x) lie in in the disk \z\ less than or equal to beta (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each beta) for which the independence polynomials have a root arbitrarily close to -beta.