{
  "id": "1301.0973",
  "version": "v1",
  "published": "2013-01-06T04:23:48.000Z",
  "updated": "2013-01-06T04:23:48.000Z",
  "title": "Which Exterior Powers are Balanced?",
  "authors": [
    "Devlin Mallory",
    "Abigail Raz",
    "Christino Tamon",
    "Thomas Zaslavsky"
  ],
  "comment": "14 pages, 2 figures",
  "journal": "Electronic Journal of Combinatorics, 20 (2013), no. 2, article P43",
  "categories": [
    "math.CO"
  ],
  "abstract": "A signed graph is a graph whose edges are given (-1,+1) weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal (-1,+1) matrix. For a signed graph $\\Sigma$ on n vertices, its exterior k-th power, where k=1,..,n-1, is a graph $\\bigwedge^{k} \\Sigma$ whose adjacency matrix is given by \\[ A({$\\bigwedge^{k} {\\Sigma}$}) = P^{\\dagger} A(\\Sigma^{\\Box k}) P, \\] where P is the projector onto the anti-symmetric subspace of the k-fold tensor product space $(\\mathbb{C}^{n})^{\\otimes k}$ and $\\Sigma^{\\Box k}$ is the k-fold Cartesian product of $\\Sigma$ with itself. The exterior power creates a signed graph from any graph, even unsigned. We prove sufficient and necessary conditions so that $\\bigwedge^{k} \\Sigma$ is balanced. For k=1,..,n-2, the condition is that either $\\Sigma$ is a signed path or $\\Sigma$ is a signed cycle that is balanced for odd k or is unbalanced for even k; for k=n-1, the condition is that each even cycle in $\\Sigma$ is positive and each odd cycle in $\\Sigma$ is negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-06T04:23:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C75"
    ],
    "keywords": [
      "signed graph",
      "adjacency matrix",
      "k-fold tensor product space",
      "exterior k-th power",
      "k-fold cartesian product"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}