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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}+"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1401.4365","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-01-07T08:09:53Z","cross_cats_sorted":[],"title_canon_sha256":"ab1723fa17e215b6307be229d70543ef7b6538c10a86c4a0ae86a9df008158ed","abstract_canon_sha256":"ccc2becdc96267c106634e6a3545ab405dbf08b475bc7c5ff8128b8c274837ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:47.989621Z","signature_b64":"w2+deMpGx7IkPmyst0lTN0LWK6efskfDqP3HISlj8BZXjAm+C8aQo5+guGPLyxp05PWuKBAo2IZ+s1N5dMW4Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ff4dbfc99773825738bad9903d6ecfed2541e0965ead5851192322162cf805a","last_reissued_at":"2026-05-18T02:55:47.988942Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:47.988942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1401.4365","created_at":"2026-05-18T02:55:47.989044+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.4365v2","created_at":"2026-05-18T02:55:47.989044+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.4365","created_at":"2026-05-18T02:55:47.989044+00:00"},{"alias_kind":"pith_short_12","alias_value":"J72NX7EZO44C","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"J72NX7EZO44CK44L","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"J72NX7EZ","created_at":"2026-05-18T12:28:33.132498+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73","json":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73.json","graph_json":"https://pith.science/api/pith-number/J72NX7EZO44CK44LVWMQHVXM73/graph.json","events_json":"https://pith.science/api/pith-number/J72NX7EZO44CK44LVWMQHVXM73/events.json","paper":"https://pith.science/paper/J72NX7EZ"},"agent_actions":{"view_html":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73","download_json":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73.json","view_paper":"https://pith.science/paper/J72NX7EZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.4365&json=true","fetch_graph":"https://pith.science/api/pith-number/J72NX7EZO44CK44LVWMQHVXM73/graph.json","fetch_events":"https://pith.science/api/pith-number/J72NX7EZO44CK44LVWMQHVXM73/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73/action/storage_attestation","attest_author":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73/action/author_attestation","sign_citation":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73/action/citation_signature","submit_replication":"https://pith.science/pith/J72NX7EZO44CK44LVWMQHVXM73/action/replication_record"}},"created_at":"2026-05-18T02:55:47.989044+00:00","updated_at":"2026-05-18T02:55:47.989044+00:00"}