{
  "id": "1506.06041",
  "version": "v1",
  "published": "2015-06-19T15:05:02.000Z",
  "updated": "2015-06-19T15:05:02.000Z",
  "title": "Alliance polynomial of regular graphs",
  "authors": [
    "Walter Carballosa",
    "José M. Rodríguez",
    "José M. Sigarreta",
    "Yadira Torres-Nuñez"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The alliance polynomial of a graph $G$ with order $n$ and maximum degree $\\Delta$ is the polynomial $A(G; x) = \\sum_{k=-\\Delta}^{\\Delta} A_{k}(G) \\, x^{n+k}$, where $A_{k}(G)$ is the number of exact defensive $k$-alliances in $G$. We obtain some properties of $A(G; x)$ and its coefficients for regular graphs. In particular, we characterize the degree of regular graphs by the number of non-zero coefficients of their alliance polynomial. Besides, we prove that the family of alliance polynomials of $\\Delta$-regular graphs with small degree is a very special one, since it does not contain alliance polynomials of graphs which are not $\\Delta$-regular. By using this last result and direct computation we find that the alliance polynomial determines uniquely each cubic graph of order less than or equal to $10$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-19T15:05:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "11B83"
    ],
    "keywords": [
      "regular graphs",
      "contain alliance polynomials",
      "alliance polynomial determines",
      "non-zero coefficients",
      "maximum degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150606041C"
    }
  }
}