{
  "abstract": "A graph is integral if the spectrum of its adjacency matrix consists entirely of integers. We prove that every simple graph having a pendant path with at least three edges has an eigenvalue in $(1,2\\cos(π/9)]$ and one in $[-2\\cos(π/9),-1)$, and hence is not integral. This settles a conjecture of Braga, Del-Vecchio and Rodrigues (2021) on integral generalized sun graphs. The argument is matrix-theoretic: adjoining a terminal path on three new coordinates to an arbitrary real symmetric matrix produces the same spectral obstruction, and the positive interval above is optimal in this generality. We then disprove a second conjecture of the same authors, which asserts that the cycle of an integral generalized sun graph other than a cycle has length divisible by four. The graph obtained from a hexagon by attaching $6,6,12,6,6$ pendant vertices to five of its six vertices is integral and has $42$ vertices. We show that it is the smallest member of an infinite family governed by the Pell equation $x^{2}-2k^{2}=-7$, and we compute in closed form the characteristic polynomial of the analogous graphs on an arbitrary even cycle. Integrality within this family forces the cycle to be a square or a hexagon, and the square case yields a second infinite family governed by the Pell equation $k^{2}-2c^{2}=1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28754",
  "author_total": 3,
  "authors": [
    {
      "name": "Rodrigo O. Braga"
    },
    {
      "name": "Jean Carlo Moraes"
    },
    {
      "name": "Matheus C. Santos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7qcvs5f63mfnx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28754",
  "fulltext_url": null,
  "id": "7qcvs5f63mfnx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28754",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7214328006",
  "pdf_url": "https://arxiv.org/pdf/2609.28754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7qcvs5f63mfnx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Pendant paths and integral generalized sun graphs",
  "title_text": "Pendant paths and integral generalized sun graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7qcvs5f63mfnx",
  "venue": "arXiv",
  "versions": []
}