Spectral Methods for Matrix Product Factorization

TL;DR

The paper investigates the spectral consequences of matrix-product factorizations $A=BC$ of graphs, linking eigenvalues, especially the spectral radius, to factors $H$ and $K$. It proves that for connected $G$, $ ho(G)= ho(H) ho(K)$, and shows how regularity and bipartiteness transfer between $G$, $H$, and $K$ under various connectivity assumptions. It also establishes connectivity implications and proves that trees (and certain forests) are not factorizable, highlighting structural constraints and parity consequences. These results deepen understanding of spectral graph factorizations and delineate clear limitations, particularly for tree-like structures and forest configurations.

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.

Figures (3)

• Figure 1: A matching of size 3 and union of two triangles
• Figure 2: The matrix product of $H$ and $K$ results the graph $G$.
• Figure 3: The picture used in the proof of \ref{['lem:A>0']}.

Theorems & Definitions (38)

• Definition 1
• Remark 1
• Lemma 1
• Theorem 2
• proof
• Remark 2
• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 6