{"paper":{"title":"Generalized Nordhaus--Gaddum Inequalities for Eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aiden Williams, Carter Antley, George Brooks, Ian Gonzalez, Joseph Aulenbacher, Linyuan Lu, Luke Hawranick, Sahil Agarwal, William Linz","submitted_at":"2026-07-17T13:24:33Z","abstract_excerpt":"For a graph $G$, let $\n\\lambda_1(G)\\ge \\lambda_2(G)\\ge \\cdots \\ge \\lambda_n(G)$ denote the adjacency eigenvalues of $G$. We investigate the asymptotic maximum of \\[ \\lambda_i(G)+\\lambda_j(\\overline G) \\] for fixed $i$ and $j$. We prove general bounds on $\\lambda_i(G) + \\lambda_{j}(\\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\\lambda_{n-i+1}(G) + \\lambda_{n-j+1}(\\overline{G})$ for fixed $i$ and $j$. We prove that for all looped graphs $G$ on $n$ vertices, \\[\\lambda_1(G) + \\lambda_2(\\overline{G}) \\le \\frac87 n. \\] Our method also gives "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15941","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.15941/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}