{
  "abstract": "For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \\leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10623",
  "author_total": 3,
  "authors": [
    {
      "name": "Jane Breen"
    },
    {
      "name": "Shaun Fallat"
    },
    {
      "name": "Johnna Parenteau"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5zvwdgkwccksp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10623",
  "fulltext_url": null,
  "id": "5zvwdgkwccksp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10623",
  "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": "W7212255851",
  "pdf_url": "https://arxiv.org/pdf/2609.10623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5zvwdgkwccksp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach",
  "title_text": "Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/5zvwdgkwccksp",
  "venue": "arXiv",
  "versions": []
}