  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 the independence
polynomial of a well-covered graph $G$ (i.e., a graph whose all maximal
independent sets are of the same size) 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.
  T. S. Michael and N. Traves (2002) provided examples of well-covered graphs
whose independence polynomials are not unimodal.
  A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain
conditions, any well-covered graph equals G* for some $G$, where G* is the
graph obtained from $G$ by appending a single pendant edge to each vertex of
$G$. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erd\"{o}s (1987) asked whether
for trees the independence polynomial is unimodal. V. E. Levit and E. Mandrescu
(2002) validated the unimodality of the independence polynomials of some
well-covered trees (e.g., $P_{n}^{*},K_{1,n}^{*}$, where $P_{n}$ is the path on
$n$ vertices and $K_{1,n}$ is the $n$-star graph).
  In this paper we show that for any graph $G$ with the stability number
alpha(G) < 5, the independence polynomial of G* is unimodal.
