On Commuting graphs of triangular rings
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929248650919936 |
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| author | Cheraghpour, Hassan Ghosseiri, Nader M. Jafari, Madineh Seyfpour, Farnaz |
| author_facet | Cheraghpour, Hassan Ghosseiri, Nader M. Jafari, Madineh Seyfpour, Farnaz |
| contents | Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $Γ(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of all $2\times 2$ upper triangular matrices over $R$ and $Γ(T)$ be the commuting graph of $T$. In this article, we find the number of edges, cliques, clique number, and independence number of $Γ(T)$ when $R$ is a finite field. Moreover, we show that for the case when $R= \mathbb{Z}_{n}$ is not a field, $Γ(T)$ is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Commuting graphs of triangular rings Cheraghpour, Hassan Ghosseiri, Nader M. Jafari, Madineh Seyfpour, Farnaz Rings and Algebras Combinatorics 16S50, 15A27, 16P10, 05C50 Let $R$ be a noncommutative ring with identity. The commuting graph of $R$, denoted by $Γ(R)$, is a graph with vertex set $R \setminus Z(R)$, and two vertices $a$, $b$ are adjacent if $a\neq b$ and $ab=ba$. Let $T=Tr(R)$ be the ring of all $2\times 2$ upper triangular matrices over $R$ and $Γ(T)$ be the commuting graph of $T$. In this article, we find the number of edges, cliques, clique number, and independence number of $Γ(T)$ when $R$ is a finite field. Moreover, we show that for the case when $R= \mathbb{Z}_{n}$ is not a field, $Γ(T)$ is connected with diameter 3. Some useful related results are also obtained, some examples are presented and a question is posed. |
| title | On Commuting graphs of triangular rings |
| topic | Rings and Algebras Combinatorics 16S50, 15A27, 16P10, 05C50 |
| url | https://arxiv.org/abs/2402.12619 |