Zero-Divisor Graphs of ℤ_n, their products and D_n

2020/09/28 by Amrita Acharyya, Acharyya, Amrita, Robinson Czajkowski +1
Computer Science · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2010.01071

openalex publication_date 2020/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is an endeavor to discuss some properties of zero-divisor graphs of the ring ℤ_n, the ring of integers modulo n. The zero divisor graph of a commutative ring R, is an undirected graph whose vertices are the nonzero zero-divisors of R, where two distinct vertices are adjacent if their product is zero. The zero divisor graph of R is denoted by Γ(R). We discussed Γ(ℤ_n)'s by the attributes of completeness, k-partite structure, complete k-partite structure, regularity, chordality, γ- β perfectness, simplicial vertices. The clique number for arbitrary Γ(ℤ_n) was also found. This work also explores related attributes of finite products Γ(ℤ_n1×⋯×ℤ_nk), seeking to extend certain results to the product rings. We find all Γ(ℤ_n1×⋯×ℤ_nk) that are perfect. Likewise, a lower bound of clique number of Γ(ℤ_m×ℤ_n) was found. Later, in this paper we discuss some properties of the zero divisor graph of the poset D_n, the set of positive divisors of a positive integer n partially ordered by divisibility.

Citations

• The Zero-Divisor Graph of a Commutative Ring
• Coloring of commutative rings

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