Determinants of Steiner Distance Hypermatrices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913839979692032 |
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| author | Cooper, Joshua Du, Zhibin |
| author_facet | Cooper, Joshua Du, Zhibin |
| contents | Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-$k$ Steiner distance hypermatrices of trees on $n$ vertices. We show that they can be nearly diagonalized as $k$-forms, generalizing a result of Graham-Lovász, implying a tensor version of ``conditional negative definiteness'', providing new proofs of previous results of the authors and Tauscheck, and resolving the conjecture that these hyperdeterminants depend only on $k$ and $n$ -- as Graham-Pollak showed for $k=2$. We conclude with some open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determinants of Steiner Distance Hypermatrices Cooper, Joshua Du, Zhibin Combinatorics 05C12 (Primary) 05C50, 15A69 (Secondary) G.2.2 Generalizing work from the 1970s on the determinants of distance hypermatrices of trees, we consider the hyperdeterminants of order-$k$ Steiner distance hypermatrices of trees on $n$ vertices. We show that they can be nearly diagonalized as $k$-forms, generalizing a result of Graham-Lovász, implying a tensor version of ``conditional negative definiteness'', providing new proofs of previous results of the authors and Tauscheck, and resolving the conjecture that these hyperdeterminants depend only on $k$ and $n$ -- as Graham-Pollak showed for $k=2$. We conclude with some open questions. |
| title | Determinants of Steiner Distance Hypermatrices |
| topic | Combinatorics 05C12 (Primary) 05C50, 15A69 (Secondary) G.2.2 |
| url | https://arxiv.org/abs/2505.10501 |