Enhanced power graphs of finite groups with cograph structure
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911222945808384 |
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| author | Bubboloni, Daniela Fumagalli, Francesco Praeger, Cheryl E. |
| author_facet | Bubboloni, Daniela Fumagalli, Francesco Praeger, Cheryl E. |
| contents | The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of $\mathcal{E}(G)$ and properties of the group $G$. We prove that if $\mathcal{E}(G)$ is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups $G$ whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with $C_4$-free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is $C_4$-free. Some open problems are posed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_18073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enhanced power graphs of finite groups with cograph structure Bubboloni, Daniela Fumagalli, Francesco Praeger, Cheryl E. Group Theory Combinatorics 05C25, 20D05 The enhanced power graph, $\mathcal{E}(G)$, of a group $G$ has vertex set $G$ and two elements are adjacent if they generate a cyclic subgroup. In the case of finite groups, we identify some striking and unexpected properties of these graphs, as well as links between properties of $\mathcal{E}(G)$ and properties of the group $G$. We prove that if $\mathcal{E}(G)$ is a cograph then it is also a chordal graph. Making use of properties of simplicial vertices, we characterise the finite groups $G$ whose enhanced power graph is diamond-free or a block graph. We also characterise the finite groups having enhanced power graph a cograph or a quasi-threshold graph, and those with $C_4$-free enhanced power graph. We use these characterisations to classify the finite nonabelian simple groups whose enhanced power graph is a cograph and give information on the finite simple groups whose enhanced power graph is $C_4$-free. Some open problems are posed. |
| title | Enhanced power graphs of finite groups with cograph structure |
| topic | Group Theory Combinatorics 05C25, 20D05 |
| url | https://arxiv.org/abs/2510.18073 |