Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917736252178432 |
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| author | Kumari, Komal Panigrahi, Pratima |
| author_facet | Kumari, Komal Panigrahi, Pratima |
| contents | The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we determine the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_19771 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups Kumari, Komal Panigrahi, Pratima Combinatorics 05C75, 05C50, 05C25 The power graph $\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \in \mathbb{N}$. Here we determine the characteristic polynomial of $\mathscr{P}(\mathbb{Z}_m \times \mathbb{Z}_{n})$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases. |
| title | Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups |
| topic | Combinatorics 05C75, 05C50, 05C25 |
| url | https://arxiv.org/abs/2407.19771 |