{
  "id": "2405.15353",
  "version": "v1",
  "published": "2024-05-24T08:45:02.000Z",
  "updated": "2024-05-24T08:45:02.000Z",
  "title": "Sharing tea on a graph",
  "authors": [
    "J. Pascal Gollin",
    "Kevin Hendrey",
    "Hao Huang",
    "Tony Huynh",
    "Bojan Mohar",
    "Sang-il Oum",
    "Ningyuan Yang",
    "Wei-Hsuan Yu",
    "Xuding Zhu"
  ],
  "comment": "19 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \\in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \\in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \\in \\mathbb{R}_{\\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\\mathbb{R}_{\\geq 0}^{V(G)}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-24T08:45:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C57",
      "05C90",
      "05C22",
      "91D30",
      "91B32",
      "05C63"
    ],
    "keywords": [
      "sharing tea",
      "arbitrary initial weight distributions",
      "finite graph",
      "social interaction",
      "consensus formation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}