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

If image denotes the number of stable sets of cardinality k in graph G, and image is the size of a maximum stable set, then image is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983) 97–106]. A graph G is very well-covered [O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177–187] if it has no isolated vertices, its order equals image and it is well-covered, i.e., all its maximal independent sets are of the same size [M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91–98]. For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by image. Under certain conditions, any well-covered graph equals image for some G [A. Finbow, B. Hartnell, R.J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. Combin. Theory Ser B 57 (1993) 44–68].