{
  "abstract": "For a graph $G$, let $ λ_1(G)\\ge λ_2(G)\\ge \\cdots \\ge λ_n(G)$ denote the adjacency eigenvalues of $G$. We prove that for all looped graphs $G$ on $n$ vertices, \\[λ_1(G) + λ_2(\\overline{G}) \\le \\frac87 n. \\] Our method also gives a short new proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that $λ_1(G) + λ_1(\\overline{G}) \\le \\frac43n - 1$. We investigate the asymptotic maximum of \\[ λ_i(G)+λ_j(\\overline G) \\] for fixed $i$ and $j$. We prove general bounds on $λ_i(G) + λ_{j}(\\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $λ_{n-i+1}(G) + λ_{n-j+1}(\\overline{G})$ for fixed $i$ and $j$. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15941",
  "author_total": 9,
  "authors": [
    {
      "name": "Sahil Agarwal"
    },
    {
      "name": "Carter Antley"
    },
    {
      "name": "Joseph Aulenbacher"
    },
    {
      "name": "George Brooks"
    },
    {
      "name": "Ian Gonzalez"
    },
    {
      "name": "Luke Hawranick"
    },
    {
      "name": "William Linz"
    },
    {
      "name": "Linyuan Lu"
    },
    {
      "name": "Aiden Williams"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4hyy0e7zd3ym5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15941",
  "fulltext_url": null,
  "id": "4hyy0e7zd3ym5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15941",
  "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": "W7169814005",
  "pdf_url": "https://arxiv.org/pdf/2607.15941",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4hyy0e7zd3ym5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2607.15941",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15941"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Generalized Nordhaus--Gaddum Inequalities for Eigenvalues",
  "title_text": "Generalized Nordhaus--Gaddum Inequalities for Eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/4hyy0e7zd3ym5",
  "venue": "arXiv",
  "versions": []
}