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

TL;DR

This work analyzes the non-zero divisor graph $\Phi(\mathbb H(\mathbb Z_{2^n}))$ attached to the Hamilton quaternions over $\mathbb Z_{2^n}$. It builds a detailed algebraic description by separating elements into units and zero-divisors, derives a universal adjacency condition via 2-adic valuations and the Smith normal form, and then investigates graph-theoretic properties. The main results show $\Phi(\mathbb H(\mathbb Z_{2^n}))$ is connected with $diam(\Phi)=2$ and contains a large unit-induced complete subgraph, while being non-planar but Hamiltonian; the paper also provides precise and bound results for the clique number $\omega$ and chromatic number $\chi$, plus an algorithm to compute neighbors and degrees for given $n$.$

Abstract

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.

Figures (2)

• Figure 1: Non-zero divisor graph of $M_2(\mathbb Z_2)$, $\Phi(M_2(\mathbb Z_2))$
• Figure 2: Non-zero divisor graph of $\mathbb H(\mathbb Z_2)$, $\Phi(\mathbb H(\mathbb Z_2))$

Theorems & Definitions (44)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Definition 2.1
• Example
• Definition 2.2
• Definition 2.3
• Remark 2.1
• Definition 2.4
• Definition 2.5