  A graph $G$ is well-covered if all its maximal stable sets have the same
size, denoted by alpha(G) (M. D. Plummer, 1970). 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 graph $G$, then it is called the independence polynomial of
$G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski
(2000) conjectured that I(G;x) is unimodal (that is, there exists an index $k$
such that the part of the sequence of coefficients from the first to $k$-th is
non-decreasing while the other part of coefficients is non-increasing) for any
well-covered graph $G$. T. S. Michael and W. N. Traves (2002) proved that this
assertion is true for alpha(G) < 4, while for alpha(G) from the set {4,5,6,7}
they provided counterexamples.
  In this paper we show that for any integer $alpha$ > 7, there exists a
(dis)connected well-covered graph $G$ with $alpha$ = alpha(G), whose
independence polynomial is not unimodal. In addition, we present a number of
sufficient conditions for a graph $G$ with alpha(G) < 7 to have unimodal
independence polynomial.
