TY - JOUR
A1 - Mönius, Katja
T1 - Eigenvalues of zero-divisor graphs of finite commutative rings
T2 - Journal of Algebraic Combinatorics
N2 - 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).
KW - EJMA-D-19-00287
KW - Zero-divisor graphs
KW - Graph eigenvalues
KW - Graphnullity
KW - Graph products
KW - Local rings
Y1 - 2021
UR - https://opus.bibliothek.uni-wuerzburg.de/frontdoor/index/index/docId/23279
UR - https://nbn-resolving.org/urn:nbn:de:bvb:20-opus-232792
SN - 0925-9899
VL - 54
ER -