{
  "id": "1801.09279",
  "version": "v1",
  "published": "2018-01-28T20:23:22.000Z",
  "updated": "2018-01-28T20:23:22.000Z",
  "title": "Topological Poincaré type inequalities and lower bounds on the infimum of the spectrum for graphs",
  "authors": [
    "Daniel Lenz",
    "Marcel Schmidt",
    "Peter Stollmann"
  ],
  "categories": [
    "math.FA",
    "math.MG"
  ],
  "abstract": "We study topological Poincar\\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-28T20:23:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower bounds",
      "study topological poincare type inequalities",
      "probability measure",
      "laplacians",
      "spectral theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}