A Note on the Values of Independence Polynomials At -1

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