Super commuting graphs of finite groups and their Zagreb indices
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917063113572352 |
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| author | Das, Shrabani Nath, Rajat Kanti |
| author_facet | Das, Shrabani Nath, Rajat Kanti |
| contents | Let $B$ be an equivalence relation defined on a finite group $G$. The $B$ super commuting graph on $G$ is a graph whose vertex set is $G$ and two distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist $g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is the $B$-equivalence class of $g \in G$. Considering $B$ as the equality, conjugacy and same order relations on $G$, in this article, we discuss the graph structures of equality/conjugacy/order super commuting graphs of certain well-known families of non-abelian groups viz. dihedral groups, dicyclic groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n}, M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{č}evi{ć} conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Super commuting graphs of finite groups and their Zagreb indices Das, Shrabani Nath, Rajat Kanti Group Theory Primary: 20D60, 20E45, Secondary: 05C25 Let $B$ be an equivalence relation defined on a finite group $G$. The $B$ super commuting graph on $G$ is a graph whose vertex set is $G$ and two distinct vertices $g$ and $h$ are adjacent if either $[g] = [h]$ or there exist $g' \in [g]$ and $h' \in [h]$ such that $g'$ commutes with $h'$, where $[g]$ is the $B$-equivalence class of $g \in G$. Considering $B$ as the equality, conjugacy and same order relations on $G$, in this article, we discuss the graph structures of equality/conjugacy/order super commuting graphs of certain well-known families of non-abelian groups viz. dihedral groups, dicyclic groups, semidihedral groups, quasidihedral groups, the groups $U_{6n}, V_{8n}, M_{2mn}$ etc. Further, we compute the Zagreb indices of these graphs and show that they satisfy Hansen-Vuki{č}evi{ć} conjecture. |
| title | Super commuting graphs of finite groups and their Zagreb indices |
| topic | Group Theory Primary: 20D60, 20E45, Secondary: 05C25 |
| url | https://arxiv.org/abs/2407.11297 |