{
  "id": "1410.8328",
  "version": "v1",
  "published": "2014-10-30T11:14:30.000Z",
  "updated": "2014-10-30T11:14:30.000Z",
  "title": "Contemplating some invariants of the Jaco Graph, $J_n(1), n \\in \\Bbb N$",
  "authors": [
    "Johan Kok",
    "Susanth C"
  ],
  "comment": "10 pages. To be submitted to the Pioneer Journal of Mathematics and Mathematical Sciences",
  "categories": [
    "math.CO"
  ],
  "abstract": "Kok et.al. [7] introduced Jaco Graphs (\\emph{order 1}). In this essay we present a recursive formula to determine the \\emph{independence number} $\\alpha(J_n(1)) = |\\Bbb I|$ with, $\\Bbb I = \\{v_{i,j}| v_1 = v_{1,1} \\in \\Bbb I$ and $v_i = v_{i,j} =v_{(d^+(v_{m, (j-1)}) + m +1)}\\}.$ We also prove that for the Jaco Graph, $J_n(1), n \\in \\Bbb N$ with the prime Jaconian vertex $v_i$ the chromatic number, $\\chi(J_n(1))$ is given by: \\begin{equation*} \\chi(J_n(1)) \\begin{cases} = (n-i) + 1, &\\text{if and only if the edge $v_iv_n$ exists,}\\\\ \\\\ = n-i &\\text{otherwise.} \\end{cases} \\end{equation*} We further our exploration in respect of \\emph{domination numbers, bondage numbers} and declare the concept of the \\emph{murtage number} of a simple connected graph $G$, denoted $m(G)$. We conclude by proving that for any Jaco Graph $J_n(1), n \\in \\Bbb N$ we have that $0 \\leq m(J_n(1)) \\leq 3.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-30T11:14:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "jaco graph",
      "invariants",
      "prime jaconian vertex",
      "chromatic number",
      "contemplating"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.8328K"
    }
  }
}