{
  "abstract": "For a graph G of order n with adjacency eigenvalues lambda_1(G) >= ... >= lambda_n(G), the cospectrality of G within the set G of all graphs of order n is the smallest spectral distance from G to another graph of G. Abdollahi and Zakeri conjectured that for every n >= 4 the cospectrality of the complete tripartite graph K_{n,n,n}, both in the Manhattan and in the Euclidean spectral distance, is attained precisely at K_{n-1,n,n+1}. We show that both conjectures are false for every n >= 4, that is, in the whole range in which they are stated. The counterexample is the graph H_n obtained from K_{n-1,n,n} by adding a vertex adjacent to the part of cardinality n - 1, and it is the natural generalization of the graph which the authors themselves exhibited for n = 3 and considered to be exceptional. We determine the spectrum of H_n in a closed form, and we show that the two spectral distances tend to 1 and to 1/2 respectively, instead of to (2*sqrt(3))/3 and to 2/3 as conjectured. On the other hand, we show that in the Euclidean distance the conjecture becomes asymptotically true when the competing graphs are restricted to complete multipartite graphs, so that it fails because a graph of rank four beats the conjectured minimizer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18636",
  "author_total": 3,
  "authors": [
    {
      "name": "Irena M. Jovanović"
    },
    {
      "name": "Jelena Sedlar"
    },
    {
      "name": "Riste Škrekovski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f4y3cxhbkjkt8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18636",
  "fulltext_url": null,
  "id": "f4y3cxhbkjkt8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18636",
  "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": "W7221750743",
  "pdf_url": "https://arxiv.org/pdf/2610.18636",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f4y3cxhbkjkt8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18636",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18636"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two conjectures on the cospectrality of complete tripartite graphs are false",
  "title_text": "Two conjectures on the cospectrality of complete tripartite graphs are false",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/f4y3cxhbkjkt8",
  "venue": "arXiv",
  "versions": []
}