  If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to
the number of stable sets of cardinality $k$ in the graph $G$, then it is
called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi,
Malde, Schwenk and Erdos (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 $a(G)$ > 3
($a(G)$ is the size of a maximum stable set of the graph $G$), while for very
well-covered graphs the conjecture is still open. In this paper we give support
to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to
${K_{1,n}, P_{n}: n > 0}$, 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).
