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

Abstract

If s k 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 ) = ∑ k = 0 α ( G ) s k x k 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 2 α ( G ) 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 G * . Under certain conditions, any well-covered graph equals G * 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]. The root of the smallest modulus of the independence polynomial of any graph is real [J.I. Brown, K. Dilcher, R.J. Nowakowski, Roots of independence polynomials of well-covered graphs, J. Algebraic Combin. 11 (2000) 197–210]. 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 / 2 α ( G * ) ) , while for a general graph of order n we show that its roots lie in | z | > 1 / ( 2 n - 1 ) . Hoede and Li [Clique polynomials and independent set polynomials of graphs, Discrete Math. 125 (1994) 219–228] posed the problem of finding graphs that can be uniquely defined by their clique polynomials ( clique-unique graphs). Stevanovic [Clique polynomials of threshold graphs, Univ. Beograd Publ. Elektrotehn. Fac., Ser. Mat. 8 (1997) 84–87] 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.