{
  "id": "1710.03249",
  "version": "v1",
  "published": "2017-10-09T18:05:29.000Z",
  "updated": "2017-10-09T18:05:29.000Z",
  "title": "Optimal Graphs for Independence and $k$-Independence Polynomials",
  "authors": [
    "J. I. Brown",
    "D. Cox"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The independence polynomial $I(G,x)$ of a finite graph $G$ is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greatest) for all non-negative $x$. Moreover, we broaden our scope to $k$-independence polynomials, which are generating functions for the $k$-clique-free subsets of vertices. For $k \\geq 3$, the results can be quite different from the $k = 2$ (i.e. independence) case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-09T18:05:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independence polynomial",
      "optimal graphs",
      "generating function",
      "independent sets",
      "finite graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}