{
  "abstract": "Koolen, Hayat and Iqbal asked for the connected graphs whose distance matrix has exactly three distinct eigenvalues, the middle one being $0$. The examples they gave, namely the Johnson graphs, the Hamming graphs and the halved cubes, are distance-regular. In this note, we show that the distance spectrum of such a graph is $\\{ρ,0,-c\\}$ with integers $c\\ge2$ and $c\\midρ$. For $c=2$ the graphs are exactly the cocktail party graphs, and for $c=3$ they are exactly the Cartesian product $K_3\\,\\square\\,K_3$ of two triangles, the Petersen graph and its complement. We also show that such a graph need not be transmission-regular: Seidel switching the lattice graph $K_6\\,\\square\\,K_6$ with respect to a line and the triangular graph $T(11)$ with respect to a maximal clique of size $10$ gives graphs on $36$ and $55$ vertices with distance spectra $\\{54,0^{(26)},(-6)^{(9)}\\}$ and $\\{81,0^{(45)},(-9)^{(9)}\\}$. Among the graphs of diameter $2$ obtained by switching a strongly regular graph with smallest eigenvalue $-2$ with respect to one part of an equitable partition, these two are the only ones that have exactly three distance eigenvalues and are not transmission-regular. We also observe that the transmission-regular graphs with exactly three distinct distance eigenvalues, the smallest one being $-3$, are exactly the complements of the strongly regular graphs with smallest eigenvalue $-2$ other than the cocktail party graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00126",
  "author_total": 1,
  "authors": [
    {
      "name": "Suil O"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8w8b874qg2ncj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00126",
  "fulltext_url": null,
  "id": "8w8b874qg2ncj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00126",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00126",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8w8b874qg2ncj.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00126",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00126"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graphs with three distinct distance eigenvalues, one of which is zero",
  "title_text": "Graphs with three distinct distance eigenvalues, one of which is zero",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/8w8b874qg2ncj",
  "venue": "arXiv",
  "versions": []
}