Partial unimodality properties of independence polynomials
نویسندگان
چکیده
A stable set in a graph G is a set of pairwise non-adjacent vertices and α(G) is the size of a maximum stable set in the graph G. The polynomial I(G; x) = s0 + s1x + s2x + ... + sαx, α = α(G), is called the independence polynomial of G (Gutman and Harary, 1983), where sk is the number of stable sets of cardinality k in G. I(G; x) is partial unimodal if the sequence of its coefficients (sk) is partial unimodal, i.e., there are some k ≤ p such that (i) s0 ≤ s1 ≤ s2 ≤ ... ≤ sk and (ii) sp ≥ sp+1 ≥ sp+2 ≥ ... ≥ sα(G). If k = p, then I(G; x) is called unimodal. In this paper, we survey the most important results referring the partial unimodality of independence polynomials of various families of graphs.
منابع مشابه
The unimodality of independence polynomials of some graphs
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems. © 2010 Elsevier Ltd. All rights reserved.
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