{
  "id": "1810.08655",
  "version": "v1",
  "published": "2018-10-19T19:11:22.000Z",
  "updated": "2018-10-19T19:11:22.000Z",
  "title": "On the roots of the subtree polynomial",
  "authors": [
    "Jason I. Brown",
    "Lucas Mol"
  ],
  "comment": "16 pages, 3 figures, comments welcome",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\\left\\{z\\in\\mathbb{C}\\colon\\ |z|\\leq 1+\\sqrt[3]{3}\\right\\}$, and that $K_{1,3}$ is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real roots of subtree polynomials contains the interval $[-2,-1]$, while the intervals $(\\infty,-1-\\sqrt[3]{3})$, $[-1,0)$, and $(0,\\infty)$ are root-free.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-19T19:11:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C31",
      "05C05"
    ],
    "keywords": [
      "subtree polynomials contains",
      "complex roots",
      "real roots",
      "collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}