On prime coprime graphs of certain finite groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912496374251520 |
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| author | Ranjan, Ravi Singh, Shubh N. |
| author_facet | Ranjan, Ravi Singh, Shubh N. |
| contents | The prime coprime graph $Θ(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate Hamiltonicity, clique number, and vertex degree of $Θ(G)$ for cyclic, dihedral, and dicyclic groups $G$. We establish that $Θ(G)$ admits a $(k,1)$-partition for cyclic, dihedral, and dicyclic groups $G$ of specified orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15993 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On prime coprime graphs of certain finite groups Ranjan, Ravi Singh, Shubh N. Group Theory 05C25, 05C76 The prime coprime graph $Θ(G)$ of a finite group $G$ is the graph whose vertex set is $G$ and any two distinct vertices are adjacent if the greatest common divisor of their orders is either $1$ or a prime. In this paper, we investigate Hamiltonicity, clique number, and vertex degree of $Θ(G)$ for cyclic, dihedral, and dicyclic groups $G$. We establish that $Θ(G)$ admits a $(k,1)$-partition for cyclic, dihedral, and dicyclic groups $G$ of specified orders. |
| title | On prime coprime graphs of certain finite groups |
| topic | Group Theory 05C25, 05C76 |
| url | https://arxiv.org/abs/2507.15993 |