The independence polynomial of a graph encapsulates all independent sets of differing sizes, a task classified as NP-hard in theoretical computer science. This article examines the independence polynomial of zero divisor graphs in commutative rings. We demonstrate that the independent sets, represented as a sequence of coefficients of the independence polynomial, exhibit unimodality and log-concavity. Therefore, for the independence polynomial of some zero divisor graphs, the unimodal conjecture is true. Additionally, the characteristics of the zeros of the independence polynomial are delineated, along with their corresponding annular regions on the plane.

翻译：图的独立多项式封装了所有不同规模的独立集，这一任务在理论计算机科学中被归类为NP难问题。本文研究了交换环中零因子图的独立多项式。我们证明了以独立多项式系数序列表示的独立集具有单峰性和对数凹性。因此，对于某些零因子图的独立多项式，单峰猜想成立。此外，还刻画了独立多项式零点的特征及其在平面上对应的环形区域。