{
  "id": "2310.00251",
  "version": "v1",
  "published": "2023-09-30T04:46:18.000Z",
  "updated": "2023-09-30T04:46:18.000Z",
  "title": "Direct sum graph of the subspaces of a finite dimensional vector space over finite fields",
  "authors": [
    "Bilal A. Wani",
    "Aaqib Altaf",
    "S. Pirzada",
    "T. A. Chishti"
  ],
  "comment": "19 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we introduce a new graph structure, called the $direct~ sum ~graph$ on a finite dimensional vector space. We investigate the connectivity, diameter and the completeness of $\\Gamma_{U\\oplus W}(\\mathbb{V})$. Further, we find its domination number and independence number. We also determine the degree of each vertex in case the base field is finite and show that the graph $\\Gamma_{U\\oplus W}(\\mathbb{V})$ is not Eulerian. We also show that under some mild conditions the graph $\\Gamma_{U\\oplus W}(\\mathbb{V})$ is triangulated. We determine the clique number of $\\Gamma_{U\\oplus W}(\\mathbb{V})$ for some particular cases. Finally, we find the size, girth, edge-connectivity and the chromatic number of $\\Gamma_{U\\oplus W}(\\mathbb{V})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-30T04:46:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C69"
    ],
    "keywords": [
      "finite dimensional vector space",
      "direct sum graph",
      "finite fields",
      "mild conditions",
      "chromatic number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}