{
  "id": "1405.2855",
  "version": "v1",
  "published": "2014-04-30T14:31:55.000Z",
  "updated": "2014-04-30T14:31:55.000Z",
  "title": "On hypergraph Lagrangians",
  "authors": [
    "Qingsong Tang",
    "Xiaojun Lu",
    "Xiangde Zhang",
    "Cheng Zhao"
  ],
  "comment": "10 pages. arXiv admin note: substantial text overlap with arXiv:1312.7529, arXiv:1211.7057, arXiv:1211.6508, arXiv:1311.1409",
  "categories": [
    "math.CO"
  ],
  "abstract": "It is conjectured by Frankl and F\\\"uredi that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges in \\cite{FF}. Motzkin and Straus' theorem confirms this conjecture when $r=2$. For $r=3$, it is shown by Talbot in \\cite{T} that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $r$-uniform hypergraphs. As an implication of this connection, we prove that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform graphs with $t$ vertices and $m$ edges satisfying ${t-1\\choose r}\\leq m \\leq {t-1\\choose r}+ {t-2\\choose r-1}-[(2r-6)\\times2^{r-1}+2^{r-3}+(r-4)(2r-7)-1]({t-2\\choose r-2}-1)$ for $r\\geq 4.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-30T14:31:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35",
      "05C65",
      "05D99",
      "90C27"
    ],
    "keywords": [
      "uniform hypergraph",
      "hypergraph lagrangians",
      "largest lagrangian",
      "conjecture",
      "colex ordering"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.2855T"
    }
  }
}