### Abstract

The independence polynomial I(G;x) of a graph G is I(G;x)=∑_{k=0} ^{α}(G)s_{k}x^{k}, where _{sk} is the number of independent sets in G of size k. The decycling number of a graph G, denoted φ(G), is the minimum size of a set SV(G) such that G-S is acyclic. Engström proved that the independence polynomial satisfies |I(G;-1)|≤^{2φ(G)} for any graph G, and this bound is best possible. Levit and Mandrescu provided an elementary proof of the bound, and in addition conjectured that for every positive integer k and integer q with |q|≤2^{k}, there is a connected graph G with φ(G)=k and I(G;-1)=q. In this note, we prove this conjecture.

Original language | English |
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Pages (from-to) | 2723-2726 |

Number of pages | 4 |

Journal | Discrete Mathematics |

Volume | 339 |

Issue number | 11 |

DOIs | |

State | Published - 6 Nov 2016 |

### Keywords

- Decycling number
- Independence polynomial

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## Cite this

Cutler, J., & Kahl, N. (2016). A note on the values of independence polynomials at -1.

*Discrete Mathematics*,*339*(11), 2723-2726. https://doi.org/10.1016/j.disc.2016.05.019