On Prime Matrix Product Factorizations

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)].