Isomorphism of Clean Graphs over $\mathbb{Z}_n$ and Structural Insight into $M_2(\mathbb{Z}_p)$
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| Format: | Preprint |
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2025
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| _version_ | 1866915495400177664 |
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| author | Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni |
| author_facet | Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni |
| contents | Let $R$ be a finite ring with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph whose vertices are pairs $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e, u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the induced subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent and } u \text{ is a unit of } R\}$. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring $M_2(\mathbb{Z}_p)$ through their $Cl_2$ graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_12004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isomorphism of Clean Graphs over $\mathbb{Z}_n$ and Structural Insight into $M_2(\mathbb{Z}_p)$ Djuang, Felicia Servina Wijayanti, Indah Emilia Susanti, Yeni Combinatorics Rings and Algebras 05C60, 05C25, 13A70, 16U60, 16U40 Let $R$ be a finite ring with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph whose vertices are pairs $(e, u)$, where $e$ is an idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e, u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The graph $Cl_2(R)$ is the induced subgraph of $Cl(R)$ induced by the set $\{(e, u): e \text{ is a nonzero idempotent and } u \text{ is a unit of } R\}$. In this study, we present properties that arise from the isomorphism of two clean graphs and conditions under which two clean graphs over direct product rings are isomorphic. We also examine the structure of the clean graph over the ring $M_2(\mathbb{Z}_p)$ through their $Cl_2$ graph. |
| title | Isomorphism of Clean Graphs over $\mathbb{Z}_n$ and Structural Insight into $M_2(\mathbb{Z}_p)$ |
| topic | Combinatorics Rings and Algebras 05C60, 05C25, 13A70, 16U60, 16U40 |
| url | https://arxiv.org/abs/2509.12004 |