## Abstract

If by s_{k} is denoted the number of independent sets of cardinality k in a graph G, then I(G; x) = s_{0} + s_{1}x +...+ s_{α}x ^{α} is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97-106, 1983), where α = α(G) is the size of a maximum independent set. The inequality {pipe}I (G; -1){pipe} ≤ 2^{ν(G)}, where ν(G) is the cyclomatic number of G, is due to (Engström in Eur. J. Comb. 30:429-438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204-1206, 2011). For ν(G) ≤ 1 it means that I(G; -1) ∈ {-2, -1, 0, 1, 2} In this paper we prove that if G is a unicyclic well-covered graph different from C_{3}, then I(G; -1) ∈ {-1, 0, 1}, while if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C_{7} or K_{2} (e. g., every well-covered tree ≠ K_{2}), then I (G; -1) = 0. Further, we demonstrate that the bounds {-2^{ν(G)}, 2^{ν(G)}} are sharp for I (G; -1), and investigate other values of I (G; -1) belonging to the interval [-2^{ν(G)}, 2^{ν(G)}].

Original language | English |
---|---|

Pages (from-to) | 259-273 |

Number of pages | 15 |

Journal | Graphs and Combinatorics |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2013 |

## Keywords

- Cyclomatic number
- Decycling number
- Independence polynomial
- Independent set
- Well-covered graph