{
  "id": "2308.12808",
  "version": "v1",
  "published": "2023-08-24T14:10:32.000Z",
  "updated": "2023-08-24T14:10:32.000Z",
  "title": "Decreasing the mean subtree order by adding $k$ edges",
  "authors": [
    "Stijn Cambie",
    "Guantao Chen",
    "Yanli Hao",
    "Nizamettin Tokar"
  ],
  "comment": "11 Pages, 5 Figures Paper identical to JGT submission",
  "categories": [
    "math.CO"
  ],
  "abstract": "The mean subtree order of a given graph $G$, denoted $\\mu(G)$, is the average number of vertices in a subtree of $G$. Let $G$ be a connected graph. Chin, Gordon, MacPhee, and Vincent [J. Graph Theory, 89(4): 413-438, 2018] conjectured that if $H$ is a proper spanning supergraph of $G$, then $\\mu(H) > \\mu(G)$. Cameron and Mol [J. Graph Theory, 96(3): 403-413, 2021] disproved this conjecture by showing that there are infinitely many pairs of graphs $H$ and $G$ with $H\\supset G$, $V(H)=V(G)$ and $|E(H)|= |E(G)|+1$ such that $\\mu(H) < \\mu(G)$. They also conjectured that for every positive integer $k$, there exists a pair of graphs $G$ and $H$ with $H\\supset G$, $V(H)=V(G)$ and $|E(H)| = |E(G)| +k$ such that $\\mu(H) < \\mu(G)$. Furthermore, they proposed that $\\mu(K_m+nK_1) < \\mu(K_{m, n})$ provided $n\\gg m$. In this note, we confirm these two conjectures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-24T14:10:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C35",
      "05C40"
    ],
    "keywords": [
      "mean subtree order",
      "graph theory",
      "average number",
      "decreasing",
      "proper spanning supergraph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}