Very well-covered graphs with log-concave independence polynomials

If sk equals the number of stable sets of cardinality k in the graph G, then I(G; x) = α(G) ∑ k=0 skx k is the independence polynomial of G (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdös (1987) conjectured that I(G; x) is unimodal whenever G is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with α(G) ≥ 4, while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if α(G) ≤ 3, or G ∈ {K1,n, Pn : n ≥ 1}, then I(G; x) is log-concave, and, hence, unimodal (where G is the very well-covered graph obtained from G by appending a single pendant edge to each vertex).