{
  "id": "2208.13462",
  "version": "v1",
  "published": "2022-08-29T10:01:51.000Z",
  "updated": "2022-08-29T10:01:51.000Z",
  "title": "Minimizers for the energy of eccentricity matrices of trees",
  "authors": [
    "Iswar Mahato",
    "M. Rajesh Kannan"
  ],
  "comment": "18 Pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The eccentricity matrix of a connected graph $G$, denoted by $\\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\\mathcal{E}(G)$ are the $\\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star. Also, we characterize the trees with minimum $\\mathcal{E}$-energy among all trees on $n$ vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-29T10:01:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eccentricity matrix",
      "minimizers",
      "eigenvalue",
      "largest nonzero entries",
      "minimum second largest"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}