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Dergiler / 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
Sayfa
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Yayın Tarihi
2021-02-04
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 Maddede Bu Makale

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Amaç & Hipotez:Let $R$ be a commutative ring with non-zero identity, and $Z(R)$ be its set of all zero-divisors.
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Temel Bulgular & Veri: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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Sonuç & Katkı:Also, we show that, for a positive integer $v$, the number of finite rings whose total graphs have vertex-arboricity (arboricity) $v$ is finite.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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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Bu Makaleye Atıf Yap

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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Kaynakça (12)

  1. D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320, 2706–2719, 2008.
  2. D.F. Anderson and A. Badawi, The total graph of a commutative ring without the zero element, J. Algebra Appl. 11 1–18 pages, 2012.
  3. G.J. Chang, C. Chen and Y. Chen, Vertex and tree arboricities of graphs, J. Comb. Optim. 8 295–306, 2004.
  4. G. Chartrand, H.V. Kronk and C.E. Wall, The point arboricity of a graph, Israel J. Math. 6, 169–175, 1968.
  5. T.T. Chelvam and T. Asir, On the genus of the total graph of a commutative ring, Comm. Algebra, 41, 142–153, 2013.
  6. B. Corbas and G.D. Williams, Ring of order p5. II. Local rings, J. Algebra, 231 (2), 691–704, 2000.
  7. H.R. Maimani, C. Wickham and S. Yassemi, Rings whose total graph have genus at most one, Rocky Mountain J. Math. 42, 1551–1560, 2012.
  8. B.R. McDonald, Finite rings with identity, Pure Appl. Math. 28, Marcel Dekker, Inc., New York, 1974.
  9. C.St.J.A. Nash-Williams, Decomposition of finite graphs into forests, Journal London Math. Soc, 39, 12, 1964.
  10. R. Raghavendran, iFinite associative rings, Compositio Math. 21, 195–229, 1969.
  11. S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discrete Math. 307, 1155–1166, 2007.
  12. B.R. McDonald, Finite rings with identity, Pure Appl. Math. 28, Marcel Dekker, Inc., New York, 1974.