{
  "abstract": "Let $G$ be a connected graph. The eccentric subgraph number $ε(v)$ of a vertex $v$ of $G$ is defined as $ε(v)=\\min\\{f_G(v,u): u\\in V(G)\\}$, where $f_G(v,u)$ is the number of connected subgraphs of $G$ containing both $v$ and $u$. The eccentric subgraph index $ES(G)$ of $G$ is defined as $ES(G)=\\displaystyle\\sum_{v\\in V(G)}ε(v).$ In this paper, we obtain the unicyclic graph that maximizes the eccentric subgraph index among all unicyclic graphs on $n$ vertices with a fixed girth. The graph that maximizes the eccentric subgraph index among all unicyclic graphs on $n$ vertices is also obtained. We then characterize the trees that maximize the eccentric subgraph index over trees on $n$ vertices with fixed diameter. Among all connected graphs on $n$ vertices with a fixed number of pendant vertices, the graph which maximizes the eccentric subgraph index is also characterized.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19694",
  "author_total": 2,
  "authors": [
    {
      "name": "Akash De"
    },
    {
      "name": "Kamal Lochan Patra"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eayhazpvq8nnr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19694",
  "fulltext_url": null,
  "id": "eayhazpvq8nnr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19694",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7221936795",
  "pdf_url": "https://arxiv.org/pdf/2610.19694",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eayhazpvq8nnr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19694",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19694"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the sum of eccentric subgraph numbers of the vertices of a connected graph",
  "title_text": "On the sum of eccentric subgraph numbers of the vertices of a connected graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/eayhazpvq8nnr",
  "venue": "arXiv",
  "versions": []
}