Spectral Methods for Matrix Product Factorization

A graph $G$ is factored into graphs $H$ and $K$ via a matrix product if there exist adjacency matrices $A$, $B$, and $C$ of $G$, $H$, and $K$, respectively, such that $A = BC$. In this paper, we study the spectral aspects of the matrix product of graphs, including regularity, bipartiteness, and connectivity. We show that if a graph $G$ is factored into a connected graph $H$ and a graph $K$ with no isolated vertices, then certain properties hold. If $H$ is non-bipartite, then $G$ is connected. If $H$ is bipartite and $G$ is not connected, then $K$ is a regular bipartite graph, and consequently, $n$ is even. Furthermore, we show that trees are not factorizable, which answers a question posed by Maghsoudi et al.