{
  "id": "2301.08106",
  "version": "v1",
  "published": "2023-01-19T14:51:49.000Z",
  "updated": "2023-01-19T14:51:49.000Z",
  "title": "Integer eigenvalues of $n$-Queens' graph",
  "authors": [
    "Domingos M. Cardoso",
    "Inês Serôdio Costa",
    "Rui Duarte"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The $n$-Queens' graph, $\\mathcal{Q}(n)$, is the graph obtained from a $n\\times n$ chessboard where each of its $n^2$ squares is a vertex and two vertices are adjacent if and only if they are in the same row, column or diagonal. In a previous work the authors have shown that, for $n\\ge4$, the least eigenvalue of $\\mathcal{Q}(n)$ is $-4$ and its multiplicity is $(n-3)^2$. In this paper we prove that $n-4$ is also an eigenvalue of $\\mathcal{Q}(n)$ and and its multiplicity is at least $\\frac{n+1}{2}$ or $\\frac{n-2}{2}$ when $n$ is odd or even, respectively. Furthermore, when $n$ is odd, it is proved that $-3,-2\\ldots,\\frac{n-11}{2}$ and $\\frac{n-5}{2},\\ldots,n-5$ are additional integer eigenvalues of $\\mathcal{Q}(n)$ and a family of eigenvectors associated with them is presented. Finally, conjectures about the the multiplicity of the aforementioned eigenvalues and about the non-existence of any other integer eigenvalue are stated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-19T14:51:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "multiplicity",
      "additional integer eigenvalues",
      "chessboard",
      "eigenvectors",
      "conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}