{
  "id": "1002.1032",
  "version": "v1",
  "published": "2010-02-04T16:20:28.000Z",
  "updated": "2010-02-04T16:20:28.000Z",
  "title": "Adjacency Matrices of Configuration Graphs",
  "authors": [
    "M. Abreu",
    "M. Funk",
    "D. Labbate",
    "V. Napolitano"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "In 1960, Hoffman and Singleton \\cite{HS60} solved a celebrated equation for square matrices of order $n$, which can be written as $$ (\\kappa - 1) I_n + J_n - A A^{\\rm T} = A$$ where $I_n$, $J_n$, and $A$ are the identity matrix, the all one matrix, and a $(0,1)$--matrix with all row and column sums equal to $\\kappa$, respectively. If $A$ is an incidence matrix of some configuration $\\cal C$ of type $n_\\kappa$, then the left-hand side $\\Theta(A):= (\\kappa - 1)I_n + J_n - A A^{\\rm T}$ is an adjacency matrix of the non--collinearity graph $\\Gamma$ of $\\cal C$. In certain situations, $\\Theta(A)$ is also an incidence matrix of some $n_\\kappa$ configuration, namely the neighbourhood geometry of $\\Gamma$ introduced by Lef\\`evre-Percsy, Percsy, and Leemans \\cite{LPPL}. The matrix operator $\\Theta$ can be reiterated and we pose the problem of solving the generalised Hoffman--Singleton equation $\\Theta^m(A)=A$. In particular, we classify all $(0,1)$--matrices $M$ with all row and column sums equal to $\\kappa$, for $\\kappa = 3,4$, which are solutions of this equation. As a by--product, we obtain characterisations for incidence matrices of the configuration $10_3F$ in Kantor's list \\cite{Kantor} and the $17_4$ configuration $#1971$ in Betten and Betten's list \\cite{BB99}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-02-04T16:20:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B30",
      "05C50"
    ],
    "keywords": [
      "adjacency matrices",
      "configuration graphs",
      "column sums equal",
      "incidence matrix",
      "matrix operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1002.1032A"
    }
  }
}