{
  "id": "2408.06992",
  "version": "v1",
  "published": "2024-08-13T15:52:36.000Z",
  "updated": "2024-08-13T15:52:36.000Z",
  "title": "On determinants of tournaments and $\\mathcal{D}_k$",
  "authors": [
    "Jing Zeng",
    "Lihua You"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $T$ be a tournament with $n$ vertices $v_1,\\ldots,v_n$. The skew-adjacency matrix of $T$ is the $n\\times n$ zero-diagonal matrix $S_T = [s_{ij}]$ in which $s_{ij}=-s_{ji}=1$ if $ v_i $ dominates $ v_j $. We define the determinant $\\det(T)$ of $ T $ as the determinant of $ S_T $. It is well-known that $\\det(T)=0$ if $n$ is odd and $\\det(T)$ is the square of an odd integer if $n$ is even. Let $\\mathcal{D}_k$ be the set of tournaments whose all subtournaments have determinant at most $ k^{2} $, where $k$ is a positive odd integer. The necessary and sufficient condition for $T\\in \\mathcal{D}_1$ or $T\\in \\mathcal{D}_3$ has been characterized in $2023$. In this paper, we characterize the set $\\mathcal{D}_5$, obtain some properties of $\\mathcal{D}_k$. Moreover, for any positive odd integer $k$, we give a construction of a tournament $T$ satisfying that $\\det(T)=k^2$, and $T\\in \\mathcal{D}_k\\backslash\\mathcal{D}_{k-2}$ if $k\\geq 3$, which implies $\\mathcal{D}_k\\backslash\\mathcal{D}_{k-2}$ is not an empty set for $k\\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-13T15:52:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C20",
      "05C50"
    ],
    "keywords": [
      "determinant",
      "positive odd integer",
      "empty set",
      "zero-diagonal matrix",
      "skew-adjacency matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}