{
  "id": "2407.10398",
  "version": "v1",
  "published": "2024-07-15T02:37:39.000Z",
  "updated": "2024-07-15T02:37:39.000Z",
  "title": "Proof of Lew's conjecture on the spectral gap of simplicial complex",
  "authors": [
    "Xiongfeng Zhan",
    "Xueyi Huang",
    "Huiqiu Lin"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $X$ be a simplicial complex on vertex set $V$ of size $n$. Let $X(k)$ denote the set of all $k$-dimensional simplices of $X$, and $\\mathrm{deg}_X(\\sigma)=|\\{\\eta\\in X(k+1):\\sigma\\subseteq \\eta\\}|$ denote the degree of $\\sigma \\in X$. A missing face in $X$ is a subset $\\sigma$ of $V$ such that $\\sigma\\notin X$ but $\\tau\\in X$ for any proper subset $\\tau$ of $\\sigma$. Let $d$ denote the maximal dimension of a missing face of $X$, and $\\mu_k(X)$ denote the $k$-th spectral gap of $X$, i.e., the smallest eigenvalue of the reduced $k$-dimensional Laplacian of $X$. In [J. Combin. Theory Ser. A 169 (2020) 105127], Lew established a lower bound for $\\mu_k(X)$: $$\\mu_k(X)\\geq (d+1)\\left(\\min_{\\sigma\\in X(k)}\\mathrm{deg}_X(\\sigma)+k+1\\right)-dn\\geq (d+1)(k+1)-dn,$$ and further conjectured that if $\\mu_k(X)=(d+1)(k+1)-dn$ for some $k$, then $X\\cong (\\Delta_d^{(d-1)})^{*(n-k-1)}*\\Delta_{(d+1)(k+1)-dn-1}$. In this paper, we confirm Lew's conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-15T02:37:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45"
    ],
    "keywords": [
      "simplicial complex",
      "confirm lews conjecture",
      "th spectral gap",
      "missing face",
      "smallest eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}