{
  "abstract": "Let $G$ be a graph of order $n$ with the adjacency eigenvalues $λ_1(G) \\geq \\dots \\geq λ_n(G) $. Let $c(v)$ denote the maximum order of a clique containing vertex $v$. We prove the vertex-localized positive square-energy inequality \\[ \\sqrt{s_+(G)} \\leq \\sum_{v\\in V}\\left(1-\\frac1{c(v)}\\right), \\] where \\[ s_+(G)=\\sum_{λ_i(G)>0}λ_i(G)^2. \\] We also characterize equality. Apart from edgeless graphs, equality holds precisely for graphs obtained from a complete regular multipartite graph by adding an arbitrary number of isolated vertices. This settles a conjecture of Kannan, Kumar and Pragada.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.24861",
  "author_total": 4,
  "authors": [
    {
      "name": "Abhay Jayarajan"
    },
    {
      "name": "M. Rajesh Kannan"
    },
    {
      "name": "Shivaramakrishna Pragada"
    },
    {
      "name": "Rahul Roy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtrwb0ddbhap2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.24861",
  "fulltext_url": null,
  "id": "dtrwb0ddbhap2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.24861",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7204257635",
  "pdf_url": "https://arxiv.org/pdf/2608.24861",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtrwb0ddbhap2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.24861",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.24861"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem",
  "title_text": "A Vertex-Localized Positive Square-Energy Strengthening of Turán's Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/dtrwb0ddbhap2",
  "venue": "arXiv",
  "versions": []
}