On the Independence Number of the Prime-Coprime Graph of a Finite Group
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908980766310400 |
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| author | Ranjan, Ravi Singh, Shubh Narayan Kumari, Surbhi Jamil, Shidra |
| author_facet | Ranjan, Ravi Singh, Shubh Narayan Kumari, Surbhi Jamil, Shidra |
| contents | The prime-coprime graph $Θ(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all finite groups $G$ for which $Θ(G)$ is a split graph. We establish a general lower bound for the independence number of $Θ(G)$ of an arbitrary finite group $G$. Moreover, we explicitly compute the independence number of $Θ(G)$ for several distinguished families of finite groups, including cyclic, dihedral, dicyclic, and semidihedral groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18475 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Independence Number of the Prime-Coprime Graph of a Finite Group Ranjan, Ravi Singh, Shubh Narayan Kumari, Surbhi Jamil, Shidra Group Theory 05C25, 05C69 The prime-coprime graph $Θ(G)$ of a finite group $G$ is the simple graph with vertex set $G$, where two distinct elements are adjacent whenever the greatest common divisor of their orders is either $1$ or a prime. We characterize all finite groups $G$ for which $Θ(G)$ is a split graph. We establish a general lower bound for the independence number of $Θ(G)$ of an arbitrary finite group $G$. Moreover, we explicitly compute the independence number of $Θ(G)$ for several distinguished families of finite groups, including cyclic, dihedral, dicyclic, and semidihedral groups. |
| title | On the Independence Number of the Prime-Coprime Graph of a Finite Group |
| topic | Group Theory 05C25, 05C69 |
| url | https://arxiv.org/abs/2604.18475 |