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)].

翻译：若图$G$的邻接矩阵$A$可分解为$A = BC$，其中$B$和$C$分别为图$H$和$K$的邻接矩阵，则称图$G$可通过矩阵乘积分解为$H$和$K$。若在任意此类分解中，总有一个因子是完美匹配（即对应置换矩阵），则称图$G$是素的。本文首先刻画所有素图，进而利用该结果对可分解森林进行完全分类，从而回答了Akbari等人[\emph{Linear Algebra and its Applications} (2025)]提出的一个问题。我们证明了所有环面图皆可分解，并完整刻画了网格图所有可能的分解方式，由此解决了Maghsoudi等人[\emph{Journal of Algebraic Combinatorics} (2025)]提出的两个问题。