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Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 50 Issue: 1

Rings whose total graphs have small vertex-arboricity and arboricity

Morteza Fatehi(Ferdowsi University of Mashhad)0000-0001-7100-666X
Kazem Khashyarmanesh(Ferdowsi University of Mashhad)0000-0002-2388-6786
Abbas Mohammadian(Ferdowsi University of Mashhad)0000-0002-2388-6786
Pages
—
Publication Date
2021-02-04
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

📌 3 Key Takeaways

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Objective & Hypothesis:Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors.
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Key Findings & Data:The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$.
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Conclusion & Impact:Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Total graph‎ · ‎arboricity‎ · vertex-arboricity

Abstract

Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors. The total graph of $R$, denoted by $T(\Gamma(R))$, is an undirected graph with all elements of $R$ as vertices, and two distinct vertices $x$ and $y$ are adjacent if and only if $x+y\in Z(R)$. In this article, we characterize, up to isomorphism, all of finite commutative rings whose total graphs have vertex-arboricity (arboricity) two or three. Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.

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Cite This Article

Fatehi, M., Khashyarmanesh, K., Mohammadian, A. (2021). Rings whose total graphs have small vertex-arboricity and arboricity. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 1. https://doi.org/10.15672/hujms.589753

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