## Eigenvalues of zero-divisor graphs of finite commutative rings

Please always quote using this URN: urn:nbn:de:bvb:20-opus-232792
• We investigate eigenvalues of the zero-divisor graph 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 Γ(R). The graph 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. We provide formulas for the nullity of Γ(R), i.e., the multiplicity of the eigenvalue 0 of Γ(R). Moreover, we precisely determine the spectra of $$\Gamma ({\mathbb {Z}}_p \times {\mathbbWe investigate eigenvalues of the zero-divisor graph 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 Γ(R). The graph 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. We provide formulas for the nullity of Γ(R), i.e., the multiplicity of the eigenvalue 0 of Γ(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)$$ for a prime number p. We introduce a graph product ×Γ with the property that Γ(R)≅Γ(R$$_1$$)×Γ⋯×ΓΓ(R$$_r$$) whenever R≅R$$_1$$×⋯×R$$_r$$. With this product, we find relations between the number of vertices of the zero-divisor graph Γ(R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Γ(R).

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Author: Katja MöniusORCiD urn:nbn:de:bvb:20-opus-232792 Journal article Fakultät für Mathematik und Informatik / Institut für Mathematik English Journal of Algebraic Combinatorics 0925-9899 2021 54 787–802 Journal of Algebraic Combinatorics 54, 787–802 (2021). https://doi.org/10.1007/s10801-020-00989-6 https://doi.org/10.1007/s10801-020-00989-6 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik EJMA-D-19-00287; Graph eigenvalues; Graph products; Graphnullity; Local rings; Zero-divisor graphs 2021/10/23 CC BY: Creative-Commons-Lizenz: Namensnennung 4.0 International