The Commuting Graph of the Quaternion Algebra over Residue Classes of Integers

Abstract

The commuting graph of an arbitrary ring $R$, denoted by $\mathrm{\Gamma }(R)$, is the graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab=ba$. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of the quaternion algebra ${\mathbb{Z}}_n[i,j,k]$.