Properties of Sub-Add Move Graphs

TL;DR

This paper introduces move graphs based on integer matrices acting on cyclic groups and investigates how their properties change with different group choices, focusing on a special class called sub-add move graphs.

Contribution

It defines move graphs on $ ext{Z}_n^m$, introduces sub-add move graphs, and analyzes their properties across different cyclic groups.

Findings

Properties vary with the choice of $ ext{Z}_n$

Characterization of sub-add move graphs

Insights into graph structure and group actions

Abstract

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 $M\!:\!\mathbb Z_n^m\to \mathbb Z_n^m$ 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.''