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) = i_kx^k. In particular, we show that if G is a well covered graph with independence number , then all the roots of i(G, x) lie in in the disk |z| (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 ) for which the independence polynomials have a root arbitrarily close to –.