{
  "abstract": "For every integer $n \\geq 11$, we construct connected, irregular, nonisomorphic graphs on $n$ vertices with equal adjacency, Laplacian, signless Laplacian, normalized Laplacian, and Seidel spectra and equal independence, clique, chromatic, complement chromatic, and maximum-cut numbers, but distinct Lovász $\\vartheta$-numbers. We join each graph in a fixed cospectral, nonisomorphic regular pair on ten vertices, due to van Dam and Haemers (2003), with $K_{n-10}$. We prove that these joins preserve equality of the five spectra and stated integer-valued invariants, and leave the respective Lovász numbers unchanged. We derive exact expressions for these numbers and prove that they are distinct. We also determine the cardinality-constrained maximum-cut profiles of both base graphs and their complements, yielding exact formulas and equality of maximum-cut numbers for the joins of either the base graphs or their complements with $K_{n-10}$. For $n=10$, we present a regular pair satisfying all stated properties except irregularity, followed by a connected, irregular, nonisomorphic pair satisfying all properties. An exhaustive computer search finds no connected, irregular, nonisomorphic pair on at most nine vertices sharing the first four spectra and all stated integer-valued invariants. Thus, ten is the smallest possible order, and such pairs exist for every $n \\geq 10$. This extends and strengthens Sason's (2024) result for \\emph{even} $n \\geq 14$, which used the neighbors splitting join operation and did not address Seidel matrices, complement chromatic numbers, or maximum-cut numbers of the graphs or joins of their complements. Thus, the Lovász number efficiently certifies nonisomorphism even when all five spectra and stated integer-valued invariants coincide.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10884",
  "author_total": 1,
  "authors": [
    {
      "name": "Igal Sason"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b360ec301e83m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10884",
  "fulltext_url": null,
  "id": "b360ec301e83m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10884",
  "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": "W7212307436",
  "pdf_url": "https://arxiv.org/pdf/2609.10884",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b360ec301e83m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2609.10884",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10884"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connected irregular cospectral graphs with distinct Lovász numbers",
  "title_text": "Connected irregular cospectral graphs with distinct Lovász numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/b360ec301e83m",
  "venue": "arXiv",
  "versions": []
}