Spectral Methods for Matrix Product Factorization
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909242967982080 |
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| author | Akbari, Saieed Fan, Yi-Zheng Hu, Fu-Tao Miraftab, Babak Wang, Yi |
| author_facet | Akbari, Saieed Fan, Yi-Zheng Hu, Fu-Tao Miraftab, Babak Wang, Yi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral Methods for Matrix Product Factorization Akbari, Saieed Fan, Yi-Zheng Hu, Fu-Tao Miraftab, Babak Wang, Yi Combinatorics Discrete Mathematics 05C50, 15A18 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. |
| title | Spectral Methods for Matrix Product Factorization |
| topic | Combinatorics Discrete Mathematics 05C50, 15A18 |
| url | https://arxiv.org/abs/2407.04150 |