Document Type

Article

Publication Date

11-6-2016

Journal / Book Title

Discrete Mathematics

Abstract

The independence polynomial I(G;x) of a graph G is I(G;x)=∑k=0 α(G)skxk, 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|≤2k, there is a connected graph G with φ(G)=k and I(G;-1)=q. In this note, we prove this conjecture.

Comments

This article is Open Access under an Elsevier User License.

DOI

10.1016/j.disc.2016.05.019

Published Citation

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

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