{
  "id": "1802.06976",
  "version": "v1",
  "published": "2018-02-20T06:09:35.000Z",
  "updated": "2018-02-20T06:09:35.000Z",
  "title": "The critical exponent: a novel graph invariant",
  "authors": [
    "Dominique Guillot",
    "Apoorva Khare",
    "Bala Rajaratnam"
  ],
  "comment": "12 pages, final version. This is an extended abstract of arXiv:1504.04069 in FPSAC 2017",
  "journal": "Seminaire Lotharingien de Combinatoire 78B (2017), Article #62",
  "categories": [
    "math.CO",
    "math.FA"
  ],
  "abstract": "A surprising result of FitzGerald and Horn (1977) shows that $A^{\\circ \\alpha} := (a_{ij}^\\alpha)$ is positive semidefinite (p.s.d.) for every entrywise nonnegative $n \\times n$ p.s.d. matrix $A = (a_{ij})$ if and only if $\\alpha$ is a positive integer or $\\alpha \\geq n-2$. Given a graph $G$, we consider the refined problem of characterizing the set $\\mathcal{H}_G$ of entrywise powers preserving positivity for matrices with a zero pattern encoded by $G$. Using algebraic and combinatorial methods, we study how the geometry of $G$ influences the set $\\mathcal{H}_G$. Our treatment provides new and exciting connections between combinatorics and analysis, and leads us to introduce and compute a new graph invariant called the critical exponent.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-20T06:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15B48"
    ],
    "keywords": [
      "novel graph invariant",
      "critical exponent",
      "combinatorial methods",
      "entrywise powers preserving positivity",
      "semidefinite"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}