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Journals / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 1

Rings whose total graphs have small vertex-arboricity and arboricity

Morteza Fatehi
Kazem Khashyarmanesh
Abbas Mohammadian
Pages
110–119
Publication Date
2022-10-18
DOI
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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. (2022). Rings whose total graphs have small vertex-arboricity and arboricity. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 1, 110–119. https://ajindex.com/articles/529456

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