Graph Powers of Groups II: The RA Matrix
Gabe Cunningham, Igor Minevich
TL;DR
This work develops a detailed, algebraic framework for graph powers G^Γ by analyzing the RA matrix $C_Γ$ and its Smith normal form to understand when a graph Γ is reducible to abelian behavior (RA) across all groups or only for specific groups. It unifies combinatorial graph structure with nonabelian group actions, showing that many graph families (notably those with girth at least 4) are RA or 1/μ-RA, and that the complexity of G^Γ is governed by the elementary divisors of $C_Γ$. By studying strong, Cartesian, and tensor graph products, the authors derive precise criteria for the resulting RA properties and demonstrate that some girth-3 graphs can realize arbitrarily large families of elementary divisors and nullity, revealing how far from RA puzzle-like behavior can be driven. The paper also provides substantial data-driven observations and constructive methods (e.g., crown graphs and Kneser graphs) to generate graphs with prescribed $C_Γ$-divisor structures, informing both theory and practical designs of nonabelian Lights Out-type puzzles.
Abstract
For a graph $Γ$ and group $G$, $G^Γ$ is the subgroup of $G^{|Γ|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $Γ$. There is a natural epimorphism $G^Γ\to (G/[G,G])^Γ$ with kernel $[G,G]^n \cap G^Γ$. When $[G,G]^n \leq G^Γ$, the structure of $G^Γ$ is easily described from $(G/[G,G])^Γ$. Fixing $Γ$, if $[G,G]^{|Γ|} \leq G^Γ$ for all $G$, we say that $Γ$ is RA (reducible to abelian). We showed in [2] that wide classes of graphs are RA, including graphs of girth 5 or more. The key tool is the RA matrix $C_Γ$, and we showed that $Γ$ is RA if and only if the row space $Row(C_Γ) = \mathbb Z^{|Γ|}$. Here, we study the possibilities for the elementary divisors of $C_Γ$; the more nontrivial elementary divisors we get, the further $Γ$ is from being RA (and the harder $G^Γ$ is to describe). We show that while many graphs, including those of girth 4, cartesian products, and most tensor products have at most one nontrivial elementary divisor, one can construct a graph of girth 3 with any prescribed set of elementary divisors and $\mathbb Z$-nullity.