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For a graph $$\Gamma $$ Γ , let K be the smallest field containing all eigenvalues of the adjacency matrix of $$\Gamma $$ Γ . The algebraic degree $$\deg (\Gamma )$$ deg ( Γ ) is the extension degree $$[K:\mathbb {Q}]$$ [ K : Q ] . In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups.

We investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$ Γ ( 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 $$\Gamma (R)$$ Γ ( R ) . The graph $$\Gamma (R)$$ Γ ( R ) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever $$xy = 0$$ x y = 0 . We provide formulas for the nullity of $$\Gamma (R)$$ Γ ( R ) , i.e., the multiplicity of the eigenvalue 0 of $$\Gamma (R)$$ Γ ( R ) . Moreover, we precisely determine the spectra of $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p ) and $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p × Z p ) for a prime number p. We introduce a graph product $$\times _{\Gamma }$$ × Γ with the property that $$\Gamma (R) \cong \Gamma (R_1) \times _{\Gamma } \cdots \times _{\Gamma } \Gamma (R_r)$$ Γ ( R ) ≅ Γ ( R 1 ) × Γ ⋯ × Γ Γ ( R r ) whenever $$R \cong R_1 \times \cdots \times R_r.$$ R ≅ R 1 × ⋯ × R r . With this product, we find relations between the number of vertices of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) , the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of $$\Gamma (R)$$ Γ ( R ) .

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