Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The independence polynomial I ( G ; x ) of a graph G is I ( G ; x ) = ź k = 0 α ( G ) s k x k , where s k is the number of independent sets in G of size k . The decycling number of a graph G , denoted ź ( G ) , is the minimum size of a set S ź V ( G ) such that G - S is acyclic. Engstrom proved that the independence polynomial satisfies | I ( G ; - 1 ) | ź 2 ź ( 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 | ź 2 k , there is a connected graph G with ź ( G ) = k and I ( G ; - 1 ) = q . In this note,  prove this conjecture.
['Jonathan Cutler', 'Nathan Kahl']
I am writing a research paper on a idea A note on the values of independence polynomials at - 1
[{"from": "human", "value": "I am writing a research paper on a idea A note on the values of independence polynomials at - 1"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The independence polynomial I ( G ; x ) of a graph G is I ( G ; x ) = \u017a k = 0 \u03b1 ( G ) s k x k , where s k is the number of independent sets in G of size k . The decycling number of a graph G , denoted \u017a ( G ) , is the minimum size of a set S \u017a V ( G ) such that G - S is acyclic. Engstrom proved that the independence polynomial satisfies | I ( G ; - 1 ) | \u017a 2 \u017a ( 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 | \u017a 2 k , there is a connected graph G with \u017a ( G ) = k and I ( G ; - 1 ) = q . In this note,  prove this conjecture."}]