{
  "id": "2209.05248",
  "version": "v1",
  "published": "2022-09-12T13:43:47.000Z",
  "updated": "2022-09-12T13:43:47.000Z",
  "title": "Inertia and spectral symmetry of eccentricity matrices of some clique trees",
  "authors": [
    "Xiaohong Li",
    "Jianfeng Wang",
    "Maurizio Brunetti"
  ],
  "comment": "16",
  "categories": [
    "math.CO"
  ],
  "abstract": "The eccentricity matrix $\\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\\mathcal C \\mathcal T$ of clique trees whose blocks have at most two cut-vertices \\textcolor{blue}{of the clique tree}. After proving the irreducibility of the eccentricity matrix of a clique tree in $\\mathcal C \\mathcal T$ and finding its inertia indices, we show that every graph in $\\mathcal C \\mathcal T$ with more than $4$ vertices and odd diameter has two positive and two negative $\\mathcal E$-eigenvalues. Positive $\\mathcal E$-eigenvalues and negative $\\mathcal E$-eigenvalues turn out to be equal in number even for graphs in $\\mathcal C \\mathcal T$ with even diameter; that shared cardinality also counts the \\textcolor{blue}{`diametrally distinguished'} vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree $G$ in $\\mathcal C \\mathcal T$ is symmetric with respect to the origin if and only if $G$ has an odd diameter and exactly two adjacent central vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-12T13:43:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C75"
    ],
    "keywords": [
      "clique tree",
      "eccentricity matrix",
      "spectral symmetry",
      "odd diameter",
      "adjacent central vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}