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arXiv · 2510.16795

On the non-zero divisor graph of the Hamilton quaternions over $\mathbb Z_{2^n}$

Gopika Govind·Chithra A. V·Manibharathi T. M. S
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Abstract

Let $R$ be a ring with unity. The non-zero divisor graph of $R$, $\Phi(R)$, 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 $\Phi(R)$ 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 $\Phi(\mathbb H(\mathbb Z_{2^n}))$, such as connectedness, adjacency of vertices, traversability, and planarity, are studied. Furthermore, we derive bounds for clique number and chromatic number.

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Gopika Govind, Chithra A. V, Manibharathi T. M. S. 2025-10-19. On the non-zero divisor graph of the Hamilton quaternions over $\mathbb Z_{2^n}$. https://arxiv.org/abs/2510.16795

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