{
  "id": "2107.03905",
  "version": "v1",
  "published": "2021-07-07T00:50:53.000Z",
  "updated": "2021-07-07T00:50:53.000Z",
  "title": "Towards a characterization of convergent sequences of $P_n$-line graphs",
  "authors": [
    "Alvaro Carbonero"
  ],
  "comment": "11 pages, 11 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $H$ and $G$ be graphs such that $H$ has at least 3 vertices and is connected. The $H$-line graph of $G$, denoted by $HL(G)$, is that graph whose vertices are the edges of $G$ and where two vertices of $HL(G)$ are adjacent if they are adjacent in $G$ and lie in a common copy of $H$. For each nonnegative integer $k$, let $HL^{k}(G)$ denote the $k$-th iteration of the $H$-line graph of $G$. We say that the sequence $\\{ HL^k(G) \\}$ converges if there exists a positive integer $N$ such that $HL^k(G) \\cong HL^{k+1}(G)$, and for $n \\geq 3$ we set $\\Lambda_n$ as the set of all graphs $G$ whose sequence $\\{HL^k(G) \\}$ converges when $H\\cong P_n$. The sets $\\Lambda_3, \\Lambda_4$ and $\\Lambda_5$ have been characterized. To progress towards the characterization of $\\Lambda_n$ in general, this paper defines and studies the following property: a graph $G$ is minimally $n$-convergent if $G\\in \\Lambda_n$ but no proper subgraph of $G$ is in $\\Lambda_n$. In addition, prove conditions that imply divergence, and use these results to develop some of the properties of minimally $n$-convergent graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-07T00:50:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C76"
    ],
    "keywords": [
      "line graph",
      "convergent sequences",
      "characterization",
      "th iteration",
      "convergent graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}