Characterizations of $p$-groups whose power graphs satisfy certain connectivity conditions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929360508813312 |
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| author | Panda, Ramesh Prasad |
| author_facet | Panda, Ramesh Prasad |
| 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 containing cycles. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} of $Γ$ is the minimum of cardinalities of the cyclic vertex cutsets of $Γ$. The {power graph} $\mathcal{P}(G)$ of a group $G$ is the undirected and simple graph whose vertices are the elements $G$ and two vertices are adjacent if one of them is the power of other in $G$. In this paper, we first characterize the finite $ p $-groups ($p$ is a prime number) whose power graphs are cyclically separable in terms of their maximal cyclic subgroups. Then we characterize the finite $ p $-groups whose power graphs have equal vertex connectivity and cyclic vertex connectivity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11809 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Characterizations of $p$-groups whose power graphs satisfy certain connectivity conditions Panda, Ramesh Prasad Group Theory 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 containing cycles. If $Γ$ has a cyclic vertex cutset, then it is said to be {cyclically separable}. The {cyclic vertex connectivity} of $Γ$ is the minimum of cardinalities of the cyclic vertex cutsets of $Γ$. The {power graph} $\mathcal{P}(G)$ of a group $G$ is the undirected and simple graph whose vertices are the elements $G$ and two vertices are adjacent if one of them is the power of other in $G$. In this paper, we first characterize the finite $ p $-groups ($p$ is a prime number) whose power graphs are cyclically separable in terms of their maximal cyclic subgroups. Then we characterize the finite $ p $-groups whose power graphs have equal vertex connectivity and cyclic vertex connectivity. |
| title | Characterizations of $p$-groups whose power graphs satisfy certain connectivity conditions |
| topic | Group Theory Combinatorics 05C25, 05C40, 20D15 |
| url | https://arxiv.org/abs/2310.11809 |