Domination and Total Domination Numbers in Zero-divisor Graphs of Commutative Rings

2025/06/03 by Sarah Anderson, Anderson, Sarah, Mike Axtell +5
Mathematics · #05C69 #13A70 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2506.02953

openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

Zero-divisor graphs of commutative rings are well-represented in the literature. In this paper, we consider dominating sets, total dominating sets, domination numbers and total domination numbers of zero-divisor graphs. We determine the domination and total domination numbers of zero-divisor graphs are equal for all zero-divisor graphs of commutative rings except for ℤ₂ × D in which D is a domain. In this case, γ(Γ(ℤ₂ × D)) = 1 and γ_t(Γ(ℤ₂ × D)) = 2.

Citations

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

Related

• On Domination in Zero-Divisor Graphs
• Domination polynomial and total domination polynomial of zero-divisor graphs of commutative rings
• Zero-Divisor Graphs of ℤ_n, their products and D_n
• The n-total graph of a commutative ring
• A Graph-Theoretic Approach to Ring Analysis: Dominant Metric Dimensions in Zero-Divisor Graphs