{
  "id": "1508.03835",
  "version": "v1",
  "published": "2015-08-16T15:13:18.000Z",
  "updated": "2015-08-16T15:13:18.000Z",
  "title": "Distance mean-regular graphs",
  "authors": [
    "V. Diego",
    "M. A. Fiol"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We introduce the concept of distance mean-regular graph, which can be seen as a generalization of both vertex-transitive and distance-regular graphs. Let $\\Gamma$ be a graph with vertex set $V$, diameter $D$, adjacency matrix $A$, and adjacency algebra ${\\cal A}$. Then, $\\Gamma$ is $distance$ $mean$-$regular$ when, for a given $u\\in V$, the averages of the intersection numbers $p_{ij}^h(u,v)=|\\Gamma_i(u)\\cap \\Gamma_j(v)|$ (number of vertices at distance $i$ from $u$ and distance $j$ from $v$) computed over all vertices $v$ at a given distance $h\\in \\{0,1,\\ldots,D\\}$ from $u$, do not depend on $u$. In this work we study some properties and characterizations of these graphs. For instance, it is shown that a distance mean-regular graph is always distance degree-regular, and we give a condition for the converse to be also true. Some algebraic and spectral properties of distance mean-regular graphs are also investigated. We show that, for distance mean regular-graphs, the role of the distance matrices of distance-regular graphs is played for the so-called distance mean-regular matrices. These matrices are computed from a sequence of orthogonal polynomials evaluated at the adjacency matrix of $\\Gamma$ and, hence, they generate a subalgebra of ${\\cal A}$. Some other algebras associated to distance mean-regular graphs are also characterized.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-16T15:13:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C50"
    ],
    "keywords": [
      "distance mean-regular graph",
      "distance-regular graphs",
      "adjacency matrix",
      "distance mean-regular matrices",
      "distance mean regular-graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}