Eigenvalues of zero-divisor graphs of finite commutative rings

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 Γ (Zp× Zp× Zp) and Γ (Zp× Zp× Zp× Zp) for a prime number p. We introduce a graph product × Γ with the property that Γ (R) ≅ Γ (R1) × Γ⋯ × ΓΓ (Rr) whenever R≅ R1× ⋯ × Rr. 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).