{
  "id": "1704.08799",
  "version": "v1",
  "published": "2017-04-28T03:40:53.000Z",
  "updated": "2017-04-28T03:40:53.000Z",
  "title": "The spectral symmetry of weakly irreducible nonnegative tensors and connected hypergraphs",
  "authors": [
    "Yi-Zheng Fan",
    "Tao Huang",
    "Yan-Hong Bao",
    "Chen-Lu Zhuan-Sun",
    "Ya-Ping Li"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\mathcal{A}$ be a weakly irreducible nonnegative tensor with spectral radius $\\rho(\\mathcal{A})$. Let $\\mathfrak{D}$ (respectively, $\\mathfrak{D}^{(0)}$) be the set of normalized diagonal matrices arising from the eigenvectors of $\\mathcal{A}$ corresponding to the eigenvalues with modulus $\\rho(\\mathcal{A})$ (respectively, the eigenvalue $\\rho(\\mathcal{A})$). Then $\\mathfrak{D}$ is a finite abelian group containing $\\mathfrak{D}^{(0)}$ as a subgroup, which acts transitively on the set $\\{e^{\\mathbf{i} \\frac{2 \\pi j}{\\ell}}\\mathcal{A}:j =0,1, \\ldots,\\ell-1\\}$, where $|\\mathfrak{D}/\\mathfrak{D}^{(0)}|=\\ell$ and $\\mathfrak{D}^{(0)}$ is the stabilizer of $\\mathcal{A}$. The spectral symmetry of $\\mathcal{A}$ is characterized by the group $\\mathfrak{D}/\\mathfrak{D}^{(0)}$, and $\\mathcal{A}$ is called spectral $\\ell$-symmetric. By analyzing the property of $\\mathfrak{D}$, we obtain some structural information of $\\mathcal{A}$ by the spectral symmetry, especially for connected hypergraphs we get some results on the edge distribution and coloring. In particular, if $\\mathcal{A}$ is also symmetric, we prove that $\\mathcal{A}$ is spectral $\\ell$-symmetric if and only if it is $(m,\\ell)$-colorable. We characterize the spectral $\\ell$-symmetry of a tensor by using its generalized traces, and show that for each positive integer $\\ell$ with $\\ell \\mid m$, there always exists an $m$-uniform hypergraph $G$ such that $G$ is spectral $\\ell$-symmetric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-28T03:40:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C65",
      "15A18",
      "15A69"
    ],
    "keywords": [
      "weakly irreducible nonnegative tensor",
      "spectral symmetry",
      "connected hypergraphs",
      "edge distribution",
      "normalized diagonal matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}