{
  "id": "1303.3457",
  "version": "v1",
  "published": "2013-03-14T14:28:28.000Z",
  "updated": "2013-03-14T14:28:28.000Z",
  "title": "Groups whose prime graphs have no triangles",
  "authors": [
    "Hung P. Tong-Viet"
  ],
  "comment": "13 pages",
  "journal": "Journal of Algebra 378 (2013), 196-206",
  "doi": "10.1016/j.jalgebra.2012.12.024",
  "categories": [
    "math.GR",
    "math.CO",
    "math.RT"
  ],
  "abstract": "Let G be a finite group and let cd(G) be the set of all complex irreducible character degrees of G Let \\rho(G) be the set of all primes which divide some character degree of G. The prime graph \\Delta(G) attached to G is a graph whose vertex set is \\rho(G) and there is an edge between two distinct primes u and v if and only if the product uv divides some character degree of G. In this paper, we show that if G is a finite group whose prime graph \\Delta(G) has no triangles, then \\Delta(G) has at most 5 vertices. We also obtain a classification of all finite graphs with 5 vertices and having no triangles which can occur as prime graphs of some finite groups. Finally, we show that the prime graph of a finite group can never be a cycle nor a tree with at least 5 vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-14T14:28:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C15",
      "05C25"
    ],
    "keywords": [
      "prime graph",
      "finite group",
      "complex irreducible character degrees",
      "product uv divides",
      "vertex set"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.3457T"
    }
  }
}