Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909754400440320 |
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| author | Dvarishkis, Blake Reynes, Josephine Rusnak, Lucas J. |
| author_facet | Dvarishkis, Blake Reynes, Josephine Rusnak, Lucas J. |
| contents | Incidence-based generalizations of cycle covers, called contributors, extend the Harary-Sachs coefficient theorem for characteristic polynomials of the adjacency matrix of graphs. All minors of the Laplacian resulting from an integer matrix are characterized using their associated oriented hypergraph through a new minimal collection of contributors to produce the coefficients of the total-minor polynomial. We prove that the natural grouping of contributors via tail-equivalence is necessarily cancellative for any contributor family that reuses an edge. We then provide a new combinatorial proof on the non-0 isospectrality of the traditional characteristic polynomials of the Laplacian and its dual. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices Dvarishkis, Blake Reynes, Josephine Rusnak, Lucas J. Combinatorics 05C50 (Primary) 05C65, 05C22, 05B20, 15A15 (Secondary) Incidence-based generalizations of cycle covers, called contributors, extend the Harary-Sachs coefficient theorem for characteristic polynomials of the adjacency matrix of graphs. All minors of the Laplacian resulting from an integer matrix are characterized using their associated oriented hypergraph through a new minimal collection of contributors to produce the coefficients of the total-minor polynomial. We prove that the natural grouping of contributors via tail-equivalence is necessarily cancellative for any contributor family that reuses an edge. We then provide a new combinatorial proof on the non-0 isospectrality of the traditional characteristic polynomials of the Laplacian and its dual. |
| title | Oriented hypergraphs and generalizing the Harary-Sachs theorem to integer matrices |
| topic | Combinatorics 05C50 (Primary) 05C65, 05C22, 05B20, 15A15 (Secondary) |
| url | https://arxiv.org/abs/2506.12271 |