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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.16795 |
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| _version_ | 1866908752084467712 |
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| author | Govind, Gopika A. V, Chithra S, Manibharathi T. M. |
| author_facet | Govind, Gopika A. V, Chithra S, Manibharathi T. M. |
| contents | Let $R$ be a ring with unity. The non-zero divisor graph of $R$, $Φ(R)$, is the graph with vertex set $R\backslash \{0,1,-1\}$, and two vertices $x$ and $y$ are adjacent if and only if either $xy$ or $yx$ is non-zero. In this article we associate $Φ(R)$ to the ring of Hamilton quaternions over $\mathbb Z_{2^n}$, $\mathbb H(\mathbb Z_{2^n})$. The detailed structure of the elements in $\mathbb H(\mathbb Z_{2^n})$ is presented, based on which various structural properties of the graph $Φ(\mathbb H(\mathbb Z_{2^n}))$, such as connectedness, adjacency of vertices, traversability, and planarity, are studied. Furthermore, we derive bounds for clique number and chromatic number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_16795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the non-zero divisor graph of the Hamilton quaternions over $\mathbb Z_{2^n}$ Govind, Gopika A. V, Chithra S, Manibharathi T. M. Combinatorics Rings and Algebras Let $R$ be a ring with unity. The non-zero divisor graph of $R$, $Φ(R)$, is the graph with vertex set $R\backslash \{0,1,-1\}$, and two vertices $x$ and $y$ are adjacent if and only if either $xy$ or $yx$ is non-zero. In this article we associate $Φ(R)$ to the ring of Hamilton quaternions over $\mathbb Z_{2^n}$, $\mathbb H(\mathbb Z_{2^n})$. The detailed structure of the elements in $\mathbb H(\mathbb Z_{2^n})$ is presented, based on which various structural properties of the graph $Φ(\mathbb H(\mathbb Z_{2^n}))$, such as connectedness, adjacency of vertices, traversability, and planarity, are studied. Furthermore, we derive bounds for clique number and chromatic number. |
| title | On the non-zero divisor graph of the Hamilton quaternions over $\mathbb Z_{2^n}$ |
| topic | Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2510.16795 |