On the numbers of independent k-sets in a claw free graph
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
Roots of Independence Polynomials of Well Covered Graphs
Journal of Algebraic Combinatorics: An International Journal
The numbers of dependent k-sets in a graph are log concave
Journal of Combinatorial Theory Series B
European Journal of Combinatorics
On the roots of independence polynomials of almost all very well-covered graphs
Discrete Applied Mathematics
The independence polynomial of rooted products of graphs
Discrete Applied Mathematics
Hi-index | 0.00 |
The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G; x) = Σk=0α(G) skxk is the independence polynomial of G (I. Gutman and F. Harary 1983). In 1990, Y.O. Hamidoune proved that for any claw-free graph G (a graph having no induced subgraph isomorphic to K1,3), I(G; x) is unimodal, i.e., there exists some k ∈ {0, 1, ..., α(G)} such that s0 ≤ s1 ≤ ... ≤ sk-1 ≤ sk ≥ sk+1 ≥ ... ≥ sα(G). Y. Alavi, P.J. Malde, A.J. Schwenk, and P. Erdös (1987) asked whether for trees the independence polynomial is unimodal. J. I. Brown, K. Dilcher and R.J. Nowakowski (2000) conjectured that I(G; x) is unimodal for any well-covered graph G (a graph whose all maximal independent sets have the same size). Michael and Traves (2002) showed that this conjecture is true for well-covered graphs with α(G) ≤ 3, and provided counterexamples for α(G) ∈ {4, 5, 6, 7}. In this paper we show that the independence polynomial of any well-covered spider is unimodal and locate its mode, where a spider is a tree having at most one vertex of degree at least three. In addition, we extend some graph transformations, first introduced in [14], respecting independence polynomials. They allow us to reduce several types of well-covered trees to claw-free graphs, and, consequently, to prove that their independence polynomials are unimodal.