On the roots of independence polynomials of almost all very well-covered graphs

If sk denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G;x) = α(G) ∑ k=0 skx k is 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 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M. D. 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 α(G∗) − α(G) for I(G∗;x). We also prove that the real roots of I(G∗;x) are in [−1,−1/2α(G∗)), while for a general graph of order n we show that its roots lie in |z| > 1/(2n− 1). Hoede and Li (1994) posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic (1997) proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders (K∗ 1,n, n ≥ 1) among well-covered trees. keywords: stable set, independence polynomial, root, well-covered graph, clique-unique graph.