{
  "id": "1701.08564",
  "version": "v1",
  "published": "2017-01-30T12:06:03.000Z",
  "updated": "2017-01-30T12:06:03.000Z",
  "title": "On sequences of polynomials arising from graph invariants",
  "authors": [
    "T. Kotek",
    "J. A. Makowsky",
    "E. V. Ravve"
  ],
  "comment": "23 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs $K_n$ and the complete bipartite graphs $K_{n,n}$ can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial $\\chi(G,X)$ of a graph $G$ evaluated at $-1$ counts the number of acyclic orientations of $G$. In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of $K_{n,n}$. We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on $n$. Finally, we discuss the distinctive power of graph polynomials in specific form.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-30T12:06:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C31",
      "05C85"
    ],
    "keywords": [
      "graph invariant",
      "polynomials arising",
      "polynomials satisfying linear recurrence relations",
      "laguerre polynomials",
      "graph polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}