In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings.

中文翻译：

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主理想环零因子图的特征值

在本文中，我们首先研究有限链环上的零因数图。我们使用缩减图确定它们的秩、行列式和特征值。此外，我们使用组合方法将工作扩展到有限交换主理想环上的零因数图，找到正特征值的数量和负特征值的数量，并找到最大特征值的上下界。最后，我们刻画了所有有限交换主理想环，使得它们的零因数图是完整的，并计算了有限交换主理想环上的零因数图的维纳指数。