  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). Let $a$ be the
size of a maximum stable set. Alavi, Malde, Schwenk and Erdos (1987)conjectured
that I(T,x) is unimodal for any tree T, while, in general, they proved that for
any permutation $p$ of {1,2,...,a} there is a graph such that
s_{p(1)}<s_{p(2)}<...<s_{p(a)}. Brown, Dilcher and Nowakowski (2000)
conjectured that I(G;x) is unimodal for any well-covered graph. Michael and
Traves (2002) provided examples of well-covered graphs with non-unimodal
independence polynomials. They proposed the "roller-coaster" conjecture: for a
well-covered graph, the subsequence (s_{a/2},s_{a/2+1},...,s_{a}) is
unconstrained in the sense of Alavi et al. The conjecture of Brown et al. is
still open for very well-covered graphs. In this paper we prove that
s_{(2a-1)/3}>=...>=s_{a-1}>=s_{a} are valid for any (a) bipartite graph $G$;
(b) quasi-regularizable graph $G$ on $2a$ vertices. In particular, we infer
that this is true for (a) trees, thus doing a step in an attempt to prove Alavi
et al.' conjecture; (b) very well-covered graphs. Consequently, for this case,
the unconstrained subsequence appearing in the roller-coaster conjecture can be
shorten to (s_{a/2},s_{a/2+1},...,s_{(2a-1)/3}). We also show that the
independence polynomial of a very well-covered graph $G$ is unimodal for a<10,
and is log-concave whenever a<6.
