Characterizing finite groups whose order supergraphs satisfy a connectivity condition
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908339352371200 |
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| author | Panda, Ramesh Prasad Ray, Papi |
| author_facet | Panda, Ramesh Prasad Ray, Papi |
| contents | Let $Γ$ be an undirected and simple graph. A set $ S $ of vertices in $Γ$ is called a {cyclic vertex cutset} of $Γ$ if $Γ- S$ is disconnected and has at least two components each containing a cycle. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. For any finite group $G$, the order supergraph $\mathcal{S}(G)$ is the simple and undirected graph whose vertices are elements of $G$, and two vertices are adjacent if as elements of $G$ the order of one divides the order of the other. In this paper, we characterize the finite nilpotent groups and various non-nilpotent groups, such as the dihedral groups, the dicyclic groups, the EPPO groups, the symmetric groups, and the alternating groups, whose order supergraphs are cyclically separable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_12307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing finite groups whose order supergraphs satisfy a connectivity condition Panda, Ramesh Prasad Ray, Papi Combinatorics 05C25, 05C40, 20D15 Let $Γ$ be an undirected and simple graph. A set $ S $ of vertices in $Γ$ is called a {cyclic vertex cutset} of $Γ$ if $Γ- S$ is disconnected and has at least two components each containing a cycle. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. For any finite group $G$, the order supergraph $\mathcal{S}(G)$ is the simple and undirected graph whose vertices are elements of $G$, and two vertices are adjacent if as elements of $G$ the order of one divides the order of the other. In this paper, we characterize the finite nilpotent groups and various non-nilpotent groups, such as the dihedral groups, the dicyclic groups, the EPPO groups, the symmetric groups, and the alternating groups, whose order supergraphs are cyclically separable. |
| title | Characterizing finite groups whose order supergraphs satisfy a connectivity condition |
| topic | Combinatorics 05C25, 05C40, 20D15 |
| url | https://arxiv.org/abs/2501.12307 |