{
  "id": "2410.13382",
  "version": "v1",
  "published": "2024-10-17T09:36:02.000Z",
  "updated": "2024-10-17T09:36:02.000Z",
  "title": "Spectra of eccentricity matrix of $H$-join of graphs",
  "authors": [
    "S. Balamoorthy",
    "T. Kavaskar"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $\\varepsilon(G)$ be the eccentricity matrix of a graph $G$ and $Spec(\\varepsilon(G))$ be the eccentricity spectrum of $G$. Let $H[G_1,G_2,\\ldots, G_k]$ be the $H$-join of graphs $G_1,G_2,\\ldots, G_k$ and let $H[G]$ be lexicographic product of $H$ and $G$. This paper finds the eccentricity matrix of a $H$-join of graphs. Using this result, we find (i) $Spec(\\varepsilon(H[G]))$ in terms of $Spec(\\varepsilon(H))$ if the radius $(rad(H))$ of $H$ is at least three; (ii) $Spec(\\varepsilon(K_k[G_1,G_2,\\ldots, G_k]))$ if $\\Delta(G_i)\\leq |V(G_i)|-2$ which generalises some of the results in \\cite{Mahato1}; (iii) $Spec(\\varepsilon(H[G_1,G_2,\\ldots, G_k]))$ if $rad(H)\\geq 2$ and $G_i$ is complete whenever $e_H(i)=2$, which generalises some of the results in \\cite{Mahato1} and \\cite{Wang1}. Finally, we find the characteristic polynomial of $\\varepsilon(K_{1,m}[G_0,G_1,\\ldots, G_m])$ if $G_i$'s are regular. As a result, we deduce some of the results in \\cite{Li}, \\cite{Mahato1}, \\cite{Patel} and \\cite{Wang}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T09:36:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C12"
    ],
    "keywords": [
      "eccentricity matrix",
      "characteristic polynomial",
      "lexicographic product",
      "generalises",
      "eccentricity spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}