{
  "id": "1508.05281",
  "version": "v1",
  "published": "2015-08-21T14:26:40.000Z",
  "updated": "2015-08-21T14:26:40.000Z",
  "title": "Directed strongly walk-regular graphs",
  "authors": [
    "Edwin R. van Dam",
    "Gholamreza Omidi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\\ell$-walk-regular with $\\ell >1$ if the number of walks of length $\\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph $\\Gamma$ with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of $\\ell$ for which $\\Gamma$ can be strongly $\\ell$-walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with nonreal eigenvalues. We give such examples and characterize those digraphs $\\Gamma$ for which there are infinitely many $\\ell$ for which $\\Gamma$ is strongly $\\ell$-walk-regular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-21T14:26:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05E30"
    ],
    "keywords": [
      "directed strongly walk-regular graphs",
      "adjacency matrix",
      "real eigenvalues resembles",
      "strongly walk-regular digraphs",
      "strong walk-regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150805281V"
    }
  }
}