  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). A graph G is very
well-covered (Favaron, 1982) if it has no isolated vertices, its order equals
2*alpha(G), where alpha(G) is the size of a maximum stable set, and it is
well-covered (i.e., all its maximal independent sets are of the same size,
Plummer, 1970). For instance, appending a single pendant edge to each vertex of
G yields a very well-covered graph, which we denote by G*. Under certain
conditions, any well-covered graph equals G* for some G (Finbow, Hartnell and
Nowakowski, 1993). The root of the smallest modulus of the independence
polynomial of any graph is real (Brown, Dilcher, and Nowakowski, 2000). The
location of the roots of the independence polynomial in the complex plane, and
the multiplicity of the root of the smallest modulus are investigated in a
number of articles. In this paper we establish formulae connecting the
coefficients of I(G;x) and I(G*;x), which allow us to show that the number of
roots of I(G;x) is equal to the number of roots of I(G*;x) different from -1,
which appears as a root of multiplicity alpha(G*)- alpha(G) for I(G*;x). We
also prove that the real roots of I(G*;x) are in [-1,-1/(2*alpha(G*)), while
for a general graph of order n we show that its roots lie in |z| > 1/(2n-1).
Using the properties of the roots of the independence polynomial, we
demonstrate that the independence polynomial distinguishes well-covered spiders
(well-covered trees with at most one vertex of degree greater than two) among
general well-covered trees.
