On finite groups whose power graphs satisfy certain connectivity conditions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912302425440256 |
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| author | Panda, Ramesh Prasad |
| author_facet | Panda, Ramesh Prasad |
| contents | Consider a graph $Γ$. 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 containing cycles. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} is the minimum cardinality of a cyclic vertex cutset of $Γ$. The power graph $\mathcal{P}(G)$ of a group $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. If $G$ is a cyclic, dihedral, or dicyclic group, we determine the order of $G$ such that $\mathcal{P}(G)$ is cyclically separable. Then we characterize the equality of vertex connectivity and cyclic vertex connectivity of $\mathcal{P}(G)$ in terms of the order of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finite groups whose power graphs satisfy certain connectivity conditions Panda, Ramesh Prasad Combinatorics Group Theory 05C25, 05C40 Consider a graph $Γ$. 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 containing cycles. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} is the minimum cardinality of a cyclic vertex cutset of $Γ$. The power graph $\mathcal{P}(G)$ of a group $G$ is the undirected simple graph with vertex set $G$ and two distinct vertices are adjacent if one of them is a positive power of the other. If $G$ is a cyclic, dihedral, or dicyclic group, we determine the order of $G$ such that $\mathcal{P}(G)$ is cyclically separable. Then we characterize the equality of vertex connectivity and cyclic vertex connectivity of $\mathcal{P}(G)$ in terms of the order of $G$. |
| title | On finite groups whose power graphs satisfy certain connectivity conditions |
| topic | Combinatorics Group Theory 05C25, 05C40 |
| url | https://arxiv.org/abs/2504.00571 |