Spectral Methods for Matrix Product Factorization

TL;DR

This paper explores the spectral properties of graph factorizations via matrix products, revealing conditions under which graph properties like connectivity and bipartiteness are preserved or implied.

Contribution

It provides new theoretical insights into the spectral characteristics of matrix product factorizations of graphs, including conditions for regularity, bipartiteness, and connectivity.

Findings

Connectedness of G when H is non-bipartite

K is regular bipartite if G is disconnected and H is bipartite

Trees are not factorizable, answering an open question

Abstract

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.