{
  "abstract": "A classical result of Andrásfai, Erdős, and Sós states that every $n$-vertex graph with odd girth at least $2k+1$ and minimum degree larger than $\\frac{2n}{2k+1}$ is bipartite. Rather than imposing a minimum-degree condition, in this paper we investigate conditions on algebraic connectivity that force graphs of given odd girth to have a simple structure. The algebraic connectivity of a graph $G$, denoted by $μ_2(G)$, is the second smallest eigenvalue of its Laplacian matrix. Our main results are as follows. 1. Every $n$-vertex triangle-free graph $G$ with $μ_2(G)\\geq \\frac{n}{3}$ is bipartite. Moreover, the constant $\\frac{1}{3}$ is asymptotically best possible. 2. For $k\\geq 3$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $μ_2(G)>\\frac{4n}{6k-1}$ is bipartite. 3. For $k\\geq 22$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $μ_2(G)>\\frac{3456n}{k^3}$ is bipartite. Moreover, the term $k^{-3}$ is asymptotically best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30799",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhengbo Chen"
    },
    {
      "name": "Chenxing Li"
    },
    {
      "name": "Zhouningxin Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vykn4c3ank03.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30799",
  "fulltext_url": null,
  "id": "3vykn4c3ank03",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30799",
  "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": "W7204964525",
  "pdf_url": "https://arxiv.org/pdf/2608.30799",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vykn4c3ank03.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30799",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30799"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the structure of graphs with given odd girth and large algebraic connectivity",
  "title_text": "On the structure of graphs with given odd girth and large algebraic connectivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/3vykn4c3ank03",
  "venue": "arXiv",
  "versions": []
}