{
  "id": "math/0403219",
  "version": "v1",
  "published": "2004-03-13T04:50:59.000Z",
  "updated": "2004-03-13T04:50:59.000Z",
  "title": "On the sandpile group of regular trees",
  "authors": [
    "Evelin Toumpakari"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following: Let T(d,h) be the d-regular tree of depth h and let V be the set of its vertices. Denote the adjacency matrix of T(d,h) by A and consider the modified Laplacian matrix D:=dI-A. Let the rows of D span the lattice L in Z^V. The sandpile group of T(d,h) is Z^V/L. We compute the rank, the exponent and the order of this abelian group and find a cyclic Hall-subgroup of order (d-1)^h. We find that the base (d-1)-logarithm of the exponent and of the order are asymptotically 3h^2/pi^2 and c_d(d-1)^h, respectively. We conjecture an explicit formula for the ranks of all Sylow subgroups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-03-13T04:50:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C25",
      "20K01",
      "82B20"
    ],
    "keywords": [
      "sandpile group",
      "regular trees",
      "abelian sandpile model",
      "d-regular tree",
      "sylow subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3219T"
    }
  }
}