{
  "abstract": "Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfying $S_k(G)<2k$. As a consequence, we prove that the path $P_n$ is the unique minimizer of $S_k(G)$ among all connected graphs of order $n$. \\vspace*{2mm} We further investigate the first Hoffman-type range \\[ 2k\\leq S_k(G)<2k+\\sqrt{2+\\sqrt5}-2. \\] We completely characterize the non-tree graphs in this range and reduce the tree case to several explicit families. The proofs combine Ky Fan's variational principle, spectral estimates from vertex-disjoint subgraphs, structural results for graphs with small spectral radius, and long-path arguments for bounded-degree graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23707",
  "author_total": 4,
  "authors": [
    {
      "name": "Shaowei Sun"
    },
    {
      "name": "Mengyao Guo"
    },
    {
      "name": "Hongyan Ge"
    },
    {
      "name": "Kinkar Chandra Das"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/01d8vs48ysp4s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23707",
  "fulltext_url": null,
  "id": "01d8vs48ysp4s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23707",
  "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": "W7214036936",
  "pdf_url": "https://arxiv.org/pdf/2609.23707",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/01d8vs48ysp4s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23707",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23707"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph",
  "title_text": "Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/01d8vs48ysp4s",
  "venue": "arXiv",
  "versions": []
}