{"paper":{"title":"More eigenvalue problems of Nordhaus-Gaddum type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vladimir Nikiforov, Xiying Yuan","submitted_at":"2014-01-07T08:09:53Z","abstract_excerpt":"Let $G$ be a graph of order $n$ and let $\\mu_{1}\\left(G\\right) \\geq \\cdots\\geq\\mu_{n}\\left(G\\right) $ be the eigenvalues of its adjacency matrix. This note studies eigenvalue problems of Nordhaus-Gaddum type. Let $\\overline{G}$ be the complement of a graph $G.$ It is shown that if $s\\geq2$ and $n\\geq15\\left(s-1\\right) ,$ then \\[ \\left\\vert \\mu_{s}\\left(G\\right) \\right\\vert +|\\mu_{s}(\\overline{G})|\\,\\leq n/\\sqrt{2\\left(s-1\\right)}-1. \\]\n  Also if $s\\geq1$ and $n\\geq4^{s},$ then \\[ \\left\\vert \\mu_{n-s+1}\\left(G\\right) \\right\\vert +|\\mu_{n-s+1}(\\overline {G})|\\,\\leq n/\\sqrt{2s}+1. \\] If $s=2^{k}+"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4365","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}