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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.10491 |
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| _version_ | 1866917102662713344 |
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| author | Singh, Randhir Patekar, S. C. |
| author_facet | Singh, Randhir Patekar, S. C. |
| contents | Let $R$ be a ring (not necessarily a commutative ring) with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph with vertices in the form of an ordered pair $(e,u)$, where $e$ is an idempotent and $u$ is a unit of ring $R$, respectively. Two distinct vertices $(e,u)$ and $(f,v)$ are adjacent in $Cl(R)$ if and only if $ef=fe=0$ or $uv=vu=1$. In this study, we considered the induced subgraph $Cl_2(R)$ of $Cl(R)$. We determined the Wiener index of $Cl_2(R)$, and we showed $Cl_2(R)$ has a perfect matching. In addition, we determined the matching number of $Cl_2(R)$ if $|U(R)|$ is not even. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_10491 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Clean Graph of a Ring Singh, Randhir Patekar, S. C. Combinatorics 05C25, 16U60, 05C12, 05C70 Let $R$ be a ring (not necessarily a commutative ring) with identity. The clean graph $Cl(R)$ of a ring $R$ is a graph with vertices in the form of an ordered pair $(e,u)$, where $e$ is an idempotent and $u$ is a unit of ring $R$, respectively. Two distinct vertices $(e,u)$ and $(f,v)$ are adjacent in $Cl(R)$ if and only if $ef=fe=0$ or $uv=vu=1$. In this study, we considered the induced subgraph $Cl_2(R)$ of $Cl(R)$. We determined the Wiener index of $Cl_2(R)$, and we showed $Cl_2(R)$ has a perfect matching. In addition, we determined the matching number of $Cl_2(R)$ if $|U(R)|$ is not even. |
| title | On the Clean Graph of a Ring |
| topic | Combinatorics 05C25, 16U60, 05C12, 05C70 |
| url | https://arxiv.org/abs/2412.10491 |