{
  "id": "2109.06269",
  "version": "v1",
  "published": "2021-09-13T19:13:01.000Z",
  "updated": "2021-09-13T19:13:01.000Z",
  "title": "A bound for the $p$-domination number of a graph in terms of its eigenvalue multiplicities",
  "authors": [
    "A. Abiad",
    "S. Akbari",
    "M. H. Fakharan",
    "A. Mehdizadeh"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected graph of order $n$ with domination number $\\gamma(G)$. Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue $\\lambda$ of $G$ with multiplicity $m_G(\\lambda)$, it holds that $\\gamma(G)\\leq n-m_G(\\lambda)$. Using techniques from the theory of star sets, in this work we prove that the same bound holds when $\\lambda$ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the $p$-domination number, and we apply it to extend the aforementioned spectral bound to the $p$-domination number using the adjacency and Laplacian eigenvalue multiplicities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-13T19:13:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "domination number",
      "multiplicity",
      "arbitrary adjacency eigenvalue",
      "laplacian eigenvalue multiplicities",
      "linear algebra appl"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}