On the Spectral Analysis of the Superpower Graph of the Direct Product of Dihedral Groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908496142794752 |
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| author | Mir, Basit Auyoob Atik, Fouzul |
| author_facet | Mir, Basit Auyoob Atik, Fouzul |
| contents | The superpower graph of a finite group $G$, or $\mathcal{S}_G$, is an undirected simple graph whose vertices are the elements of the group $G$, and two distinct vertices $a,b\in G$ are adjacent if and only if the order of one vertex divides the order of the other vertex, which means that either $o(a)|o(b)$ or $o(b)|o(a)$. In this paper, we have investigated the $A_α$-adjacency spectral properties of the superpower graph of the direct product $D_p\times D_p$, where $D_p$ is a dihedral group for $p$ being prime. Also, we have determined its Laplacian and signless Laplacian spectrum by giving different values to $α$; furthermore, we delved into its superpower graph and deduced the $A_α$- adjacency spectrum of the superpower graph of $D_p\times D_p$ and $D_{p^m}$ for $p$ being an odd prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Spectral Analysis of the Superpower Graph of the Direct Product of Dihedral Groups Mir, Basit Auyoob Atik, Fouzul Spectral Theory 05C50, 05C25 The superpower graph of a finite group $G$, or $\mathcal{S}_G$, is an undirected simple graph whose vertices are the elements of the group $G$, and two distinct vertices $a,b\in G$ are adjacent if and only if the order of one vertex divides the order of the other vertex, which means that either $o(a)|o(b)$ or $o(b)|o(a)$. In this paper, we have investigated the $A_α$-adjacency spectral properties of the superpower graph of the direct product $D_p\times D_p$, where $D_p$ is a dihedral group for $p$ being prime. Also, we have determined its Laplacian and signless Laplacian spectrum by giving different values to $α$; furthermore, we delved into its superpower graph and deduced the $A_α$- adjacency spectrum of the superpower graph of $D_p\times D_p$ and $D_{p^m}$ for $p$ being an odd prime. |
| title | On the Spectral Analysis of the Superpower Graph of the Direct Product of Dihedral Groups |
| topic | Spectral Theory 05C50, 05C25 |
| url | https://arxiv.org/abs/2508.06812 |