The independence polynomial of a graph G is \(I(G;x)=\sum _{k=0}^{\alpha (G)} s_{k}\cdot x^{k}\), where \(s_{k}\) and \(\alpha (G)\) denote the number of independent sets of cardinality k and the independence number of G, respectively. We say that a cycle is a \(\tilde{3}\)-cycle if its length is divisible by 3, otherwise a non-\(\tilde{3}\)-cycle. Define \(\phi (G)\) to be the decycling number of G. Engström proved that \(|I(G;-1)| \le 2^{\phi (G)}\) for any graph G. In this paper, we first prove that \(|I(G;-1)|\le 2^{\beta (G)}-\beta (G)\) for graphs with non-\(\tilde{3}\)-cycles, where \(\beta (G)\) is the cyclomatic number of G. Infinitely many examples show that there do exist graphs satisfying \(\beta (G)=\phi (G)\) and containing non-\(\tilde{3}\)-cycles. In such a case, it improves the Engström’s result. Furthermore, \(|I(G;-1)|\le 2^{k}\) provided that all cycles of G are pairwise disjoint, where k is the number of \(\tilde{3}\)-cycles of G. This provides a new perspective on estimating the independence polynomial at -1 of many special graphs. In the case G contains no vertices of degree 1, \(|I(G;-1)|\le 2^{\beta (G)}-1\) if \(\beta (G)\ge 2\), and \(|I(G;-1)|\le 2^{\beta (G)-1}\) if G contains a non-\(\tilde{3}\)-cycle.

图G的独立多项式为\（I（G; x）= \ sum _ {k = 0} ^ {\ alpha（G）} s_ {k} \ cdot x ^ {k} \），其中\（ s_ {k} \）和\（\ alpha（G）\）分别表示基数k的独立集合数和G的独立数。我们说，如果一个循环的长度可以被3整除，那么它就是一个\（\ tilde {3} \）循环，否则为非\（\ tilde {3} \）循环。将\（\ phi（G）\）定义为G的循环编号。Engström证明了任何图G的\（| I（G; -1）| \ le 2 ^ {\ phi（G）} \）。在本文中，我们首先证明\（| I（G; -1）| \ le 2 ^ {\ beta（G）}-\ beta（G）\）对于具有非\（\ tilde {3} \）-圈的图，其中\（ \ beta（G）\）是G的圈数。无限多的示例表明，确实存在满足\（\ beta（G）= \ phi（G）\）且包含非\（\ tilde {3} \）-圈的图。在这种情况下，它可以改善Engström的效果。此外，\（| I（G; -1）| \ le 2 ^ {k} \）假设G的所有周期成对不相交，其中k是\（\ tilde {3} \）-的周期数摹。这为估计许多特殊图的-1处的独立多项式提供了新的视角。在这种情况下如果\（\ beta（G）\ ge 2 \）和\\，则G不包含1度的顶点\\ ||（I（G; -1）| \ le 2 ^ {\ beta（G）}-1 \）。 （| I（G; -1）| \ le 2 ^ {\ beta（G）-1} \），如果G包含非\（\ tilde {3} \）-循环。