{
  "abstract": "We confirm a conjecture of Brause, Randerath, Rautenbach and Schiermeyer (2016) by proving a localized lower bound on the independence number of a graph that strengthens the classical bounds of Fajtlowicz (1978) and of Caro (1979) and Wei (1981), which in turn settles a conjecture by Bertram and Horák (1996). Our proof is based on a new Motzkin--Straus-type inequality involving local clique numbers and the independence number. We then apply the developed methods to study spectral and algebraic measures of graph bipartiteness. In particular, we extend a theorem of Brandt (1998) on spectral bipartiteness from regular $K_{r+1}$-free graphs to all $K_{r+1}$-free graphs, we improve a general upper bound for the least signless Laplacian eigenvalue of $K_{r+1}$-free graphs, and we disprove a conjecture of de Lima, Nikiforov and Oliveira (2016) in the case of $K_4$-free graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00210",
  "author_total": 3,
  "authors": [
    {
      "name": "Aida Abiad"
    },
    {
      "name": "Hitesh Kumar"
    },
    {
      "name": "Shivaramakrishna Pragada"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/50vqy88z42syh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00210",
  "fulltext_url": null,
  "id": "50vqy88z42syh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00210",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7206191636",
  "pdf_url": "https://arxiv.org/pdf/2609.00210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/50vqy88z42syh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Localization of the Caro-Wei bound and its applications to bipartiteness",
  "title_text": "Localization of the Caro-Wei bound and its applications to bipartiteness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/50vqy88z42syh",
  "venue": "arXiv",
  "versions": []
}