  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 non-zero 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 $\Gamma(R) \cong \Gamma(R_1)
\times_{\Gamma} \ldots \times_{\Gamma} \Gamma(R_r)$ whenever $R \cong R_1
\times \ldots \times R_r.$ 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
$\Gamma(R)$.

Combinatorics;Commutative Algebra;Rings and Algebras