On Prime Matrix Product Factorizations

TL;DR

This paper characterizes prime graphs based on matrix factorizations of their adjacency matrices, classifies factorable forests, and explores factorizations of tori and grids, answering open questions in graph theory and algebraic combinatorics.

Contribution

It provides a complete characterization of prime graphs and classifies factorable forests, also analyzing factorizations of tori and grids.

Findings

All prime graphs are characterized.

Factorable forests are classified.

Every torus is factorable, with detailed grid factorizations.

Abstract

A graph $G$ factors into graphs $H$ and $K$ via a matrix product if $A = BC$, where $A$, $B$, and $C$ are the adjacency matrices of $G$, $H$, and $K$, respectively. The graph $G$ is prime if, in every such factorization, one of the factors is a perfect matching that is, it corresponds to a permutation matrix. We characterize all prime graphs, then using this result we classify all factorable forests, answering a question of Akbari et al. [\emph{Linear Algebra and its Applications} (2025)]. We prove that every torus is factorable, and we characterize all possible factorizations of grids, addressing two questions posed by Maghsoudi et al. [\emph{Journal of Algebraic Combinatorics} (2025)].