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Graph Powers of Groups

Gabe Cunningham, Igor Minevich

TL;DR

This work generalizes Lights Out by allowing vertex states to lie in a finite group $G$ and studying the resulting graph power $G^\Gamma$, with a focus on when $\Gamma$ is reducible to abelian analysis (RA). The authors develop a lattice/SNF-based framework to understand $G^\Gamma$ for abelian $G$ and extend it to nonabelian $G$ via commutator subgroups, introducing RA, commutator hierarchies, and the RA Matrix $C_\Gamma$. A central result is that $\Gamma$ is RA if and only if the RA Matrix has full rank over $\mathbb{Z}$ (or equivalently modulo every prime for Heisenberg groups), with many graph families (e.g., Petersen, trees, complete bipartite graphs under gcd conditions, even-dimensional cubes) shown to be RA, while specific graphs like odd-dimensional cubes $Q_{2n+1}$ and folded cubes $\square_d$ (under stated conditions) are not. The framework provides concrete computational tools (e.g., GAP/Sage code) to decide RA and reveals a rich interaction between graph topology, group structure, and linear-algebraic invariants, offering a pathway toward broader graph-product analyses of group actions on coupled systems.

Abstract

The Lights Out Puzzle, played on a graph $Γ$, has been studied using linear algebra over $\mathbb{F}_2$ and more generally over $\mathbb{Z}/k\mathbb{Z}$. We generalize the setting by allowing the states of vertices to be the elements of a group $G$, where a \textit{click} in vertex $v$ multiplies the state of $v$ and its neighbors by an element $g \in G$ on the right. Starting with the identity element $e \in G$ for all vertices, the totality of all achievable state configurations forms a group $G^Γ$. This group generalizes parallel products of group actions and provides a rich structure for analysis. For many graphs, which we term ``RA'' (reducible to abelian), the problem reduces -- regardless of $G$ -- to a linear algebra question over $\mathbb{Z}$. We discuss a chain of five different subgroups consisting of commutators and introduce techniques for showing that families of graphs are RA using each. In particular, using Heisenberg groups, we establish that a graph is RA precisely when a certain lattice spans $\mathbb{Z}^{|Γ|}$. While most graphs appear to be RA, we show the odd-dimensional cube graphs $Q_{2n+1}$ and folded cube graphs $\square_d$, for $d$ odd or 2, are not.

Graph Powers of Groups

TL;DR

This work generalizes Lights Out by allowing vertex states to lie in a finite group and studying the resulting graph power , with a focus on when is reducible to abelian analysis (RA). The authors develop a lattice/SNF-based framework to understand for abelian and extend it to nonabelian via commutator subgroups, introducing RA, commutator hierarchies, and the RA Matrix . A central result is that is RA if and only if the RA Matrix has full rank over (or equivalently modulo every prime for Heisenberg groups), with many graph families (e.g., Petersen, trees, complete bipartite graphs under gcd conditions, even-dimensional cubes) shown to be RA, while specific graphs like odd-dimensional cubes and folded cubes (under stated conditions) are not. The framework provides concrete computational tools (e.g., GAP/Sage code) to decide RA and reveals a rich interaction between graph topology, group structure, and linear-algebraic invariants, offering a pathway toward broader graph-product analyses of group actions on coupled systems.

Abstract

The Lights Out Puzzle, played on a graph , has been studied using linear algebra over and more generally over . We generalize the setting by allowing the states of vertices to be the elements of a group , where a \textit{click} in vertex multiplies the state of and its neighbors by an element on the right. Starting with the identity element for all vertices, the totality of all achievable state configurations forms a group . This group generalizes parallel products of group actions and provides a rich structure for analysis. For many graphs, which we term ``RA'' (reducible to abelian), the problem reduces -- regardless of -- to a linear algebra question over . We discuss a chain of five different subgroups consisting of commutators and introduce techniques for showing that families of graphs are RA using each. In particular, using Heisenberg groups, we establish that a graph is RA precisely when a certain lattice spans . While most graphs appear to be RA, we show the odd-dimensional cube graphs and folded cube graphs , for odd or 2, are not.

Paper Structure

This paper contains 22 sections, 36 theorems, 38 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graph theory
  4. Group theory
  5. Linear algebra
  6. Graph powers of groups
  7. Definition and examples
  8. Subgroups of $G$ and $G^S$
  9. How the structure of $\Gamma$ affects $G^\Gamma$
  10. Graph powers of an abelian group
  11. Determining $G^S$ from the lattice $\mathcal{L} = \langle S\rangle$
  12. Description of $\mathbb{Z}^\Gamma$ for some basic graphs
  13. The elementary divisors of small graphs
  14. Reducing nonabelian graph powers to the abelian case
  15. Definitions
  16. ...and 7 more sections

Key Result

Proposition 2.1

Suppose $M$ is $m \times n$ with $m \ge n$. If there is a prime $p$ such that every row sum of $M$ is divisible by $p$, then $M$ has an elementary divisor that is divisible by $p$ (possibly equal to 0).

Figures (3)

  • Figure 1: GAP code to compute $G^\Gamma$
  • Figure 2: GAP code to show that $Q_3$ is $D_{10}$-RA but not $D_8$-RA.
  • Figure 3: SageMath code to check all graphs with up to 7 vertices for the RA property.

Theorems & Definitions (85)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • ...and 75 more