{
  "id": "1605.08665",
  "version": "v1",
  "published": "2016-05-27T14:32:41.000Z",
  "updated": "2016-05-27T14:32:41.000Z",
  "title": "Combinatorial methods for the spectral p-norm of hypermatrices",
  "authors": [
    "V. Nikiforov"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The spectral $p$-norm of $r$-matrices generalizes the spectral $2$-norm of $2$-matrices. In 1911 Schur gave an upper bound on the spectral $2$-norm of $2$-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to $r$-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for $2$-matrices. The main result of this paper extends the latter result to $r$-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on new combinatorial concepts like $r$\\emph{-partite }% $r$\\emph{-matrix} and \\emph{symmetrant} of a matrix, which appear to be instrumental in the study of the spectral $p$-norm in general. Thus, another application shows that the spectral $p$-norm and the $p$-spectral radius of a symmetric nonnegative $r$-matrix are equal whenever $p\\geq r$. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the $p$-spectral radius and the spectral $p$-norm of $r$-matrices and $r$-graphs.\\medskip",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-27T14:32:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C65",
      "15A18",
      "15A42",
      "15A60",
      "15A69"
    ],
    "keywords": [
      "spectral p-norm",
      "combinatorial methods",
      "hypermatrices",
      "spectral radius",
      "combinatorial concepts"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}