{
  "id": "2405.09093",
  "version": "v1",
  "published": "2024-05-15T05:01:44.000Z",
  "updated": "2024-05-15T05:01:44.000Z",
  "title": "Line graphs and Nordhaus-Gaddum-type bounds for self-loop graphs",
  "authors": [
    "Saieed Akbari",
    "Irena M. Jovanović",
    "Johnny Lim"
  ],
  "comment": "19 pages. To appear in Bulletin of the Malaysian Mathematical Sciences Society",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G_S$ be the graph obtained by attaching a self-loop at every vertex in $S \\subseteq V(G)$ of a simple graph $G$ of order $n.$ In this paper, we explore several new results related to the line graph $L(G_S)$ of $G_S.$ Particularly, we show that every eigenvalue of $L(G_S)$ must be at least $-2,$ and relate the characteristic polynomial of the line graph $L(G)$ of $G$ with the characteristic polynomial of the line graph $L(\\widehat{G})$ of a self-loop graph $\\widehat{G}$, which is obtained by attaching a self-loop at each vertex of $G$. Then, we provide some new bounds for the eigenvalues and energy of $G_S.$ As one of the consequences, we obtain that the energy of a connected regular complete multipartite graph is not greater than the energy of the corresponding self-loop graph. Lastly, we establish a lower bound of the spectral radius in terms of the first Zagreb index $M_1(G)$ and the minimum degree $\\delta(G),$ as well as proving two Nordhaus-Gaddum-type bounds for the spectral radius and the energy of $G_S,$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-15T05:01:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C90",
      "05C92"
    ],
    "keywords": [
      "line graph",
      "self-loop graph",
      "nordhaus-gaddum-type bounds",
      "connected regular complete multipartite graph",
      "characteristic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}