Properties of Sub-Add Move Graphs

We introduce the notion of a move graph, that is, a directed graph whose vertex set is a $\mathbb{Z}$-module ${\mathbb{Z}}_n^m$, and whose arc set is uniquely determined by the action $M\text{\hspace{0.17em}⁣}:\text{\hspace{0.17em}⁣}{\mathbb{Z}}_n^m\to {\mathbb{Z}}_n^m$ where $M$ is an $m\times m$ matrix with integer entries. We study the manner in which properties of move graphs differ when one varies the choice of cyclic group ${\mathbb{Z}}_n$. Our principal focus is on a special family of such graphs, which we refer to as ``sub-add move graphs.‘’