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Properties of Sub-Add Move Graphs

Patrick Cesarz, Eugene Fiorini, Charles Gong, Kyle Kelley, Philip Thomas, Andrew Woldar

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.''

Properties of Sub-Add Move Graphs

Abstract

We introduce the notion of a move graph, that is, a directed graph whose vertex set is a -module , and whose arc set is uniquely determined by the action where is an matrix with integer entries. We study the manner in which properties of move graphs differ when one varies the choice of cyclic group . Our principal focus is on a special family of such graphs, which we refer to as ``sub-add move graphs.''

Paper Structure

This paper contains 7 sections, 19 theorems, 13 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The general case
  4. The sub-add move graph
  5. The sub-add move graph $\Gamma_{M,\,n}$, $n=2^r$
  6. The sub-add move graph $\Gamma_{M,\,n}$, $n$ odd
  7. The sub-add move graph $\Gamma_{M,\,p}$, $p$ an odd prime

Key Result

Theorem 3.1

Let $M \in {\rm Mat}_{m}(\mathbb{Z})$ have $\mathbb Z_n$-order $k$. Suppose $M^{t}{\bf x}^T={\bf x}^T$ for some ${\bf x}\in V(\Gamma_{M,\,n})$, $t\in\mathbb N$. Then ${\bf x}$ is on a unique directed cycle in $\Gamma_{M,\,n}$ of length dividing $t$. In particular, all directed cycles in $\Gamma_{M,\

Figures (5)

  • Figure 1: The move graph $\Gamma_{M,\,3}$ of Example \ref{['example:n=3']}.
  • Figure 2: The sub-add move graph $\Gamma_{M,\,3}$.
  • Figure 3: The sub-add move graph $\Gamma_{M,\,4}$.
  • Figure 4: The sub-add move graph $\Gamma_{M,\,5}$ illustrates a special case of Theorem \ref{['thm:n-odd']}.
  • Figure 5: The sub-add move graph $\Gamma_{M,\,6}$ illustrates a special case of Theorem \ref{['thm:(2L+1)2^k']}.

Theorems & Definitions (42)

  • Example 2.1
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 32 more