@article{Moenius2021,
  author    = {M{\"o}nius, Katja},
  title     = {Eigenvalues of zero-divisor graphs of finite commutative rings},
  series = {Journal of Algebraic Combinatorics},
  volume    = {54},
  journal   = {Journal of Algebraic Combinatorics},
  issn      = {0925-9899},
  doi       = {10.1007/s10801-020-00989-6},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-232792},
  pages     = {787-802},
  year      = {2021},
  abstract  = {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 \(\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)\) 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).},
  language  = {en}
}