## 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).…

Author: | Katja MöniusORCiD |
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URN: | urn:nbn:de:bvb:20-opus-232792 |

Document Type: | Journal article |

Faculties: | Fakultät für Mathematik und Informatik / Institut für Mathematik |

Language: | English |

Parent Title (English): | Journal of Algebraic Combinatorics |

ISSN: | 0925-9899 |

Year of Completion: | 2021 |

Volume: | 54 |

Pagenumber: | 787–802 |

Source: | Journal of Algebraic Combinatorics 54, 787–802 (2021). https://doi.org/10.1007/s10801-020-00989-6 |

DOI: | https://doi.org/10.1007/s10801-020-00989-6 |

Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

Tag: | EJMA-D-19-00287; Graph eigenvalues; Graph products; Graphnullity; Local rings; Zero-divisor graphs |

Release Date: | 2021/10/23 |

Licence (German): | CC BY: Creative-Commons-Lizenz: Namensnennung 4.0 International |