Discrete Mathematics
On the numbers of independent k-sets in a claw free graph
Journal of Combinatorial Theory Series B
A characterization of well covered graphs of girth 5 or greater
Journal of Combinatorial Theory Series B
Clique polynomials and independent set polynomials of graphs
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
Clique polynomials have a unique root of smallest modulus
Information Processing Letters
Roots of Independence Polynomials of Well Covered Graphs
Journal of Algebraic Combinatorics: An International Journal
The independence fractal of a graph
Journal of Combinatorial Theory Series B
On the Location of Roots of Independence Polynomials
Journal of Algebraic Combinatorics: An International Journal
The roots of the independence polynomial of a clawfree graph
Journal of Combinatorial Theory Series B
On unimodality of independence polynomials of some well-covered trees
DMTCS'03 Proceedings of the 4th international conference on Discrete mathematics and theoretical computer science
The independence polynomial of rooted products of graphs
Discrete Applied Mathematics
Journal of Combinatorial Optimization
Independence polynomials of some compound graphs
Discrete Applied Mathematics
Cohen-Macaulay Graphs and Face Vectors of Flag Complexes
SIAM Journal on Discrete Mathematics
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If s"k denotes the number of stable sets of cardinality k in graph G, and @a(G) is the size of a maximum stable set, then I(G;x)=@?"k"="0^@a^(^G^)s"kx^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@a(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 @a(G^*)-@a(G) for I(G^*;x). We also prove that the real roots of I(G^*;x) are in [-1,-1/2@a(G^*)), while for a general graph of order n we show that its roots lie in |z|1/(2n-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.