The independence polynomial of a graph at -1

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G;x) = s0 + s1x + ... + sαx α is the independence polynomial of G [12], where α = α(G) is the size of a maximum stable set. In this paper we prove that I(G;−1) satisfies |I(G;−1)| ≤ 2, where ν(G) equals the cyclomatic number of G, and the bounds are sharp. In particular, if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C7 or K2 (e.g., a well-covered tree 6= K2), then I(G;−1) = 0.