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

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.