Characterization of commuting graphs of finite groups having small genus
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909256125513728 |
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| author | Das, Shrabani Nongsiang, Deiborlang Nath, Rajat Kanti |
| author_facet | Das, Shrabani Nongsiang, Deiborlang Nath, Rajat Kanti |
| contents | In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only $K_8 \sqcup 9K_1$, $K_8 \sqcup 5K_2$, $K_8 \sqcup 3K_4$, $K_8 \sqcup 9K_3$, $K_8\sqcup 9(K_1 \vee 3K_2)$, $3K_6$ and $3K_6 \sqcup 4K_4 \sqcup 6K_2$ can be realized as commuting graphs of finite groups. As consequences of our results we also show that for any finite non-abelian group $G$ if the commuting graph of $G$ (denoted by $Γ_c(G)$) is double-toroidal or triple-toroidal then $Γ_c(G)$ and its complement satisfy Hansen-Vuki{č}evi{ć} Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8$, that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of commuting graphs of finite groups having small genus Das, Shrabani Nongsiang, Deiborlang Nath, Rajat Kanti Group Theory In this paper we first show that among all double-toroidal and triple-toroidal finite graphs only $K_8 \sqcup 9K_1$, $K_8 \sqcup 5K_2$, $K_8 \sqcup 3K_4$, $K_8 \sqcup 9K_3$, $K_8\sqcup 9(K_1 \vee 3K_2)$, $3K_6$ and $3K_6 \sqcup 4K_4 \sqcup 6K_2$ can be realized as commuting graphs of finite groups. As consequences of our results we also show that for any finite non-abelian group $G$ if the commuting graph of $G$ (denoted by $Γ_c(G)$) is double-toroidal or triple-toroidal then $Γ_c(G)$ and its complement satisfy Hansen-Vuki{č}evi{ć} Conjecture and E-LE conjecture. In the process we find a non-complete graph, namely the non-commuting graph of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3) \rtimes Q_8$, that is hyperenergetic. This gives a new counter example to a conjecture of Gutman regarding hyperenergetic graphs. |
| title | Characterization of commuting graphs of finite groups having small genus |
| topic | Group Theory |
| url | https://arxiv.org/abs/2401.00993 |