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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.09697 |
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| _version_ | 1866911917969244160 |
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| author | Klanderman, Sarah Montee, MurphyKate Piotrowski, Andrzej Rice, Alex Shader, Bryan |
| author_facet | Klanderman, Sarah Montee, MurphyKate Piotrowski, Andrzej Rice, Alex Shader, Bryan |
| contents | The Seidel matrix of a tournament on $n$ players is an $n\times n$ skew-symmetric matrix with entries in $\{0, 1, -1\}$ that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an $n\times n$ Seidel matrix is $0$ if $n$ is odd, and is an odd perfect square if $n$ is even. This leads to the study of the set \[ \mathcal{D}(n)= \{ \sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. \] This paper studies various questions about $\mathcal{D}(n)$. It is shown that $\mathcal{D}(n)$ is a proper subset of $\mathcal{D}(n+2)$ for every positive even integer, and every odd integer in the interval $[1, 1+n^2/2]$ is in $\mathcal{D}(n)$ for $n$ even. The expected value and variance of $\det S$ over the $n\times n$ Seidel matrices chosen uniformly at random is determined, and upper bounds on $\max \mathcal{D}(n)$ are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many $n$, $\mathcal{D}(n)$ contains a gap (that is, there are odd integers $k<\ell <m$ such that $k, m \in \mathcal{D}(n)$ but $\ell \notin \mathcal{D}(n)$) and several properties of the characteristic polynomials of Seidel matrices are established. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_09697 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Determinants of Seidel Tournament Matrices Klanderman, Sarah Montee, MurphyKate Piotrowski, Andrzej Rice, Alex Shader, Bryan Combinatorics 53C20, 15A15 The Seidel matrix of a tournament on $n$ players is an $n\times n$ skew-symmetric matrix with entries in $\{0, 1, -1\}$ that encapsulates the outcomes of the games in the given tournament. It is known that the determinant of an $n\times n$ Seidel matrix is $0$ if $n$ is odd, and is an odd perfect square if $n$ is even. This leads to the study of the set \[ \mathcal{D}(n)= \{ \sqrt{\det S}: \mbox{ $S$ is an $n\times n$ Seidel matrix}\}. \] This paper studies various questions about $\mathcal{D}(n)$. It is shown that $\mathcal{D}(n)$ is a proper subset of $\mathcal{D}(n+2)$ for every positive even integer, and every odd integer in the interval $[1, 1+n^2/2]$ is in $\mathcal{D}(n)$ for $n$ even. The expected value and variance of $\det S$ over the $n\times n$ Seidel matrices chosen uniformly at random is determined, and upper bounds on $\max \mathcal{D}(n)$ are given, and related to the Hadamard conjecture. Finally, it is shown that for infinitely many $n$, $\mathcal{D}(n)$ contains a gap (that is, there are odd integers $k<\ell <m$ such that $k, m \in \mathcal{D}(n)$ but $\ell \notin \mathcal{D}(n)$) and several properties of the characteristic polynomials of Seidel matrices are established. |
| title | Determinants of Seidel Tournament Matrices |
| topic | Combinatorics 53C20, 15A15 |
| url | https://arxiv.org/abs/2406.09697 |