A Family of Well-Covered Graphs with Unimodal Independence Polynomials
نویسنده
چکیده
T. S. Michael and N. Traves (2002) provided examples of wellcovered graphs whose independence polynomials are not unimodal. A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G∗ for some G, where G∗ is the graph obtained from G by appending a single pendant edge to each vertex of G. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees the independence polynomial is unimodal. V. E. Levit and E. Mandrescu (2002) validated the unimodality of the independence polynomials of some well-covered trees (e.g., P ∗ n ,K ∗ 1,n, where Pn is the path on n vertices and K1,n is the n-star graph). In this paper we show that for any graph G with α(G) ≤ 4, the independence polynomial of G∗ is unimodal.
منابع مشابه
On the unimodality of independence polynomials of very well-covered graphs
The independence polynomial i(G, x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing...
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