{
  "abstract": "We prove that the triangle $C_3$ is the only integral unicyclic graph with an odd cycle. The main tool is a spectral inequality that does not require integrality: if $G$ is a connected non-bipartite graph and $G-e$ is bipartite for some edge $e$, then the eigenvalues $λ_1\\ge\\cdots\\geλ_n$ of $G$ satisfy $\\sum_{i}(λ_i+λ_{n+1-i})^+\\le 2$ and $0<λ_1+λ_n\\le 1$. It follows from comparing $A(G)$ with the signed adjacency matrix obtained by reversing the sign of $e$, which is similar to $-A(G)$, through a common rank-one perturbation and Weyl's monotonicity theorem. For integral graphs in this class, an arithmetic lemma then gives $λ_n=1-λ_1$ and at least $λ_1-1\\ge1$ triangles, which yields the classification. We also prove that an integral graph of odd girth $g$ has at least $(g-1)!/2$ cycles of length $g$, with equality for the Petersen graph when $g=5$, and we derive consequences for quantum periodicity a",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22773",
  "author_total": 3,
  "authors": [
    {
      "name": "Rodrigo O. Braga"
    },
    {
      "name": "Jean Carlo Moraes"
    },
    {
      "name": "Matheus C. Santos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bf0yvd36cc043.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22773",
  "fulltext_url": null,
  "id": "bf0yvd36cc043",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22773",
  "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": "W7222964904",
  "pdf_url": "https://arxiv.org/pdf/2610.22773",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bf0yvd36cc043.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22773",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22773"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Spectral asymmetry and odd cycles in integral graphs",
  "title_text": "Spectral asymmetry and odd cycles in integral graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/bf0yvd36cc043",
  "venue": "arXiv",
  "versions": []
}