{
  "id": "1807.01551",
  "version": "v1",
  "published": "2018-07-04T13:00:43.000Z",
  "updated": "2018-07-04T13:00:43.000Z",
  "title": "Spectral gaps, missing faces and minimal degrees",
  "authors": [
    "Alan Lew"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $\\sigma\\notin X$ such that $\\tau\\in X$ for any $\\tau\\subsetneq \\sigma$. For a $k$-dimensional simplex $\\sigma$ in $X$, its degree in $X$ is the number of $(k+1)$-dimensional simplices in $X$ containing it. Let $\\delta_k$ denote the minimal degree of a $k$-dimensional simplex in $X$. Let $L_k$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $\\mu_k(X)$ denote its minimal eigenvalue. We prove the following lower bound on the spectral gaps $\\mu_k(X)$, for complexes $X$ without missing faces of dimension larger than $d$: \\[ \\mu_k(X)\\geq (d+1)(\\delta_k+k+1)-d n. \\] As a consequence we obtain a new proof of a vanishing result for the homology of simplicial complexes without large missing faces. We present a family of examples achieving equality at all dimensions, showing that the bound is tight. For $d=1$ we characterize the equality case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-04T13:00:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral gaps",
      "minimal degree",
      "dimensional simplex",
      "simplicial complex",
      "equality case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}