{
  "id": "1510.00924",
  "version": "v1",
  "published": "2015-10-04T11:17:02.000Z",
  "updated": "2015-10-04T11:17:02.000Z",
  "title": "Characterizing $2$-Distance Graphs and Solving the Equations $T_2(X)=kP_2$ or $K_m \\cup K_n$",
  "authors": [
    "Ramuel P. Ching",
    "I. J. L. Garces"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $X$ be a finite, simple graph with vertex set $V(X)$. The $2$-distance graph $T_2(X)$ of $X$ is the graph with the same vertex set as $X$ and two vertices are adjacent if and only if their distance in $X$ is exactly $2$. A graph $G$ is a $2$-distance graph if there exists a graph $X$ such that $T_2(X)=G$. In this paper, we give three characterizations of $2$-distance graphs, and find all graphs $X$ such that $T_2(X)=kP_2$ or $K_m \\cup K_n$, where $k \\ge 2$ is an integer, $P_2$ is the path of order $2$, and $K_m$ is the complete graph of order $m \\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-04T11:17:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12"
    ],
    "keywords": [
      "distance graph",
      "vertex set",
      "characterizing",
      "simple graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}