{
  "id": "2309.01358",
  "version": "v1",
  "published": "2023-09-04T04:48:05.000Z",
  "updated": "2023-09-04T04:48:05.000Z",
  "title": "Inertia and spectral symmetry of the eccentricity matrices of a class of bi-block graphs",
  "authors": [
    "T. Divyadevi",
    "I. Jeyaraman"
  ],
  "comment": "26 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The eccentricity matrix of a simple connected graph G is obtained from the distance matrix of G by retaining the largest non-zero distance in each row and column, and the remaining entries are defined to be zero. A bi-block graph is a simple connected graph whose blocks are all complete bipartite graphs with possibly different orders. In this paper, we study the eccentricity matrices of a subclass B (which includes trees) of bi-block graphs. We first find the inertia of the eccentricity matrices of graphs in B, and thereby we characterize graphs in B with odd diameters. Precisely, if G in B with diameter of G greater than three, then we show that the eigenvalues of the eccentricity matrix of G are symmetric with respect to the origin if and only if the diameter of G is odd. Further, we prove that the eccentricity matrices of graphs in B are irreducible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-04T04:48:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C50"
    ],
    "keywords": [
      "eccentricity matrix",
      "bi-block graph",
      "spectral symmetry",
      "simple connected graph",
      "complete bipartite graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}