{
  "id": "1506.06819",
  "version": "v1",
  "published": "2015-06-22T23:37:00.000Z",
  "updated": "2015-06-22T23:37:00.000Z",
  "title": "Simplicial and Cellular Trees",
  "authors": [
    "Art M. Duval",
    "Caroline J. Klivans",
    "Jeremy L. Martin"
  ],
  "comment": "39 pages (including 5-page bibliography); 5 figures. Chapter for forthcoming IMA volume \"Recent Trends in Combinatorics\"",
  "categories": [
    "math.CO"
  ],
  "abstract": "Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai and Adin, and more recently by numerous authors, the fundamental topological properties of a tree --- namely acyclicity and connectedness --- can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley's formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-22T23:37:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E45"
    ],
    "keywords": [
      "cellular trees",
      "simplicial",
      "algebraic graph theory",
      "yields higher-dimensional analogues",
      "cellular homology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606819D"
    }
  }
}