We investigate eigenvalues of the zero-divisor graph \(\Gamma (R)\) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of \(\Gamma (R)\). The graph \(\Gamma (R)\) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever \(xy = 0\). We provide formulas for the nullity of \(\Gamma (R)\), i.e., the multiplicity of the eigenvalue 0 of \(\Gamma (R)\). Moreover, we precisely determine the spectra of \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) and \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) for a prime number p. We introduce a graph product \(\times _{\Gamma }\) with the property that \(\Gamma (R) \cong \Gamma (R_1) \times _{\Gamma } \cdots \times _{\Gamma } \Gamma (R_r)\) whenever \(R \cong R_1 \times \cdots \times R_r.\) With this product, we find relations between the number of vertices of the zero-divisor graph \(\Gamma (R)\), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of \(\Gamma (R)\).

我们研究有限交换环R的零除图\（\ Gamma（R）\）的特征值，并研究这些特征值，R的环理论性质和\（\ Gamma（ R）\）。图\（\ Gamma（R）\）定义为具有顶点集的图，该顶点集由R的所有非零零除数和每当\（xy = 0 \）的相邻顶点x，  y组成。我们对的无效提供式\（\伽玛（R）\） ，即特征值0的的多重\（\伽玛（R）\） 。而且，我们精确地确定了\（\ Gamma（{\ mathbb {Z}} _ p \ times {\ mathbb {Z}} _ p \ times {\ mathbb {Z}} _ p）\）和\（\ Gamma（{\ mathbb {Z}} _ p \ times {\ mathbb {Z}} _ p \ times {\ mathbb {Z}} _ p \ times {\ mathbb {Z}} _ p）\）质数p。我们引入具有\（\ Gamma（R）\ cong \ Gamma（R_1）\ times _ {\ Gamma} \ cdots \ times _ {\ Gamma}的属性的图形产品 \（\ times _ {\ Gamma} \）\ Gamma（R_r）\）每当\（R \ cong R_1 \ times \ cdots \ times R_r。\）使用此乘积，我们发现零因数图的顶点数\（\ Gamma（R）\ ），压缩的零因子图，环R的结构和\（\ Gamma（R）\）的特征值。