Abstract
The independence polynomialI(G;x) of a graph G is I(G;x)=∑k=0α(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 S⊆V(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.
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