Eigenvalues of zero-divisor graphs of finite commutative rings

2019/10/28 by Mönius, Katja
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1910.12567

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 non-zero 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 Γ(\mathbb Z_p × \mathbb Z_p × \mathbb Z_p) and Γ(\mathbb Z_p × \mathbb Z_p × \mathbb Z_p × \mathbb Z_p) for a prime number p. We introduce a graph product ×_Γ with the property that Γ(R) ≅ Γ(R₁) ×_Γ … ×_Γ Γ(R_r) whenever R ≅ R₁ × … × 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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