## Abstract

If sk equals the number of stable sets of cardinality k in the graph G, then I(G; x) = ^{σα (G)} _{k=0} skxk is the independence polynomial of G (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdös (1987) conjectured that I(G; x) is unimodal whenever G is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that I(G; x) is unimodal for any well- covered graph G. Michael and Traves (2002) showed that the assertion is false for well-covered graphs with (G) ≤ 4, while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if α(G) ≤ 3, or G {K1,n, Pn : n ≥ 1}, then I(G* x) is log-concave, and, hence, unimodal (where G* is the very well-covered graph obtained from G by appending a single pendant edge to each vertex).

Original language | English |
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Pages (from-to) | 73-80 |

Number of pages | 8 |

Journal | Carpathian Journal of Mathematics |

Volume | 20 |

Issue number | 1 |

State | Published - 2004 |

Externally published | Yes |

## Keywords

- Claw-free graph
- Independence polynomial
- Log-concavity
- Stable set
- Tree
- Unimodality
- Well-covered graph