{
  "id": "2107.09162",
  "version": "v1",
  "published": "2021-07-19T21:21:33.000Z",
  "updated": "2021-07-19T21:21:33.000Z",
  "title": "On a conjecture of Laplacian energy of trees",
  "authors": [
    "Hilal A. Ganiea",
    "Bilal A. Rather",
    "S. Pirzada"
  ],
  "categories": [
    "math.CO",
    "math.SP"
  ],
  "abstract": "Let $G$ be a simple graph with $n$ vertices, $m$ edges having Laplacian eigenvalues $\\mu_1, \\mu_2, \\dots, \\mu_{n-1},\\mu_n=0$. The Laplacian energy $LE(G)$ is defined as $LE(G)=\\sum_{i=1}^{n}|\\mu_i-\\overline{d}|$, where $\\overline{d}=\\frac{2m}{n}$ is the average degree of $G$. Radenkovi\\'{c} and Gutman conjectured that among all trees of order $n$, the path graph $P_n$ has the smallest Laplacian energy. Let $ \\mathcal{T}_{n}(d) $ be the family of trees of order $n$ having diameter $ d $. In this paper, we show that Laplacian energy of any tree $T\\in \\mathcal{T}_{n}(4)$ is greater than the Laplacian energy of $P_n$, thereby proving the conjecture for all trees of diameter $4$. We also show the truth of conjecture for all trees with number of non-pendent vertices at most $\\frac{9n}{25}-2$. Further, we give some sufficient conditions for the conjecture to hold for a tree of order $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-19T21:21:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C12",
      "15A18"
    ],
    "keywords": [
      "conjecture",
      "smallest laplacian energy",
      "average degree",
      "laplacian eigenvalues",
      "path graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}