{
  "id": "1905.09017",
  "version": "v1",
  "published": "2019-05-22T08:35:51.000Z",
  "updated": "2019-05-22T08:35:51.000Z",
  "title": "On the resistance distance and Kirchhoff index of a linear hexagonal (cylinder) chain",
  "authors": [
    "Sumin Huang",
    "Shuchao Li"
  ],
  "comment": "20pages, 15 figures,",
  "categories": [
    "math.CO"
  ],
  "abstract": "The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$\\, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yields the linear hexagonal cylinder chain, written as $R_n$. We obtain explicit formulae for the resistance distance $r_{L_n}(i, j)$ (resp. $r_{R_n}(i,j)$) between any two vertices $i$ and $j$ of $L_n$ (resp. $R_n$). To the best of our knowledge $\\{L_n\\}_{n=1}^{\\infty}$ and $\\{R_n\\}_{n=1}^{\\infty}$ are two nontrivial families with diameter going to $\\infty$ for which all resistance distances have been explicitly calculated. We determine the maximum and the minimum resistance distances in $L_n$ (resp. $R_n$). The monotonicity and some asymptotic properties of resistance distances in $L_n$ and $R_n$ are given. As well we give formulae for the Kirchhoff indices of $L_n$ and $R_n$ respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-22T08:35:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kirchhoff index",
      "linear hexagonal cylinder chain",
      "minimum resistance distances",
      "linear hexagonal chain",
      "opposite lateral edges"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}