Spectral decomposition of hypergraph automorphism compatible matrices
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913339512193024 |
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| author | Banerjee, Anirban Parui, Samiron |
| author_facet | Banerjee, Anirban Parui, Samiron |
| contents | This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an \( f \)-compatible matrices capture aspects of the symmetry, represented by \( f \), within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any \(f\)-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising \( f \). Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral decomposition of hypergraph automorphism compatible matrices Banerjee, Anirban Parui, Samiron Combinatorics 05C65, 05C50, 05C07, 05C30 This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an \( f \)-compatible matrices capture aspects of the symmetry, represented by \( f \), within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any \(f\)-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising \( f \). Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices. |
| title | Spectral decomposition of hypergraph automorphism compatible matrices |
| topic | Combinatorics 05C65, 05C50, 05C07, 05C30 |
| url | https://arxiv.org/abs/2405.01364 |