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Cioab\\u{a}, Desai and Tait proved this conjecture for $n\\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $\\gamma=\\gamma(k)$ such that the same conclusion holds for $n\\ge \\Omega(k^{\\gamma(k)})$.\n  We prove a stronger uniform theorem for Nikiforov"},"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":"2606.06987","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-05T07:25:27Z","cross_cats_sorted":[],"title_canon_sha256":"219120e4dad3a2d8bd83e44b61a85a919ba24b6a71a6d1b517a6da161927f4a3","abstract_canon_sha256":"52f893c67961a15a3a4a3475f473be238604177a6f785dc834e242512df8500e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:39.714054Z","signature_b64":"p4+Ze7O0hENVWGKO9fOXjyzRcN/SUEVQIrp/79g0ZMOVisJcKYEfB/GmBuK7jcJeOXMnvQVPtBj8Qajch0WsBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"25ff7379555ae3fa4f53fadba06ab016ee71dcfa44dbdd7c561e6623eae36a84","last_reissued_at":"2026-06-08T01:04:39.713243Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:39.713243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight Bound for Nikiforov's Spectral Even-Cycle Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Feng Liu, Jiasheng Zeng, Peiru Kuang, Shuang Sun, Yan Wang","submitted_at":"2026-06-05T07:25:27Z","abstract_excerpt":"Nikiforov conjectured that, for every fixed $k\\ge2$ and all sufficiently large $n$, the unique $n$-vertex $C_{2k+2}$-free graph with maximum adjacency spectral radius is $S^+_{n,k}$, where $S_{n,k}=K_k\\vee\\overline K_{n-k}$ and $S^+_{n,k}$ is obtained from $S_{n,k}$ by adding one edge inside the independent part. Cioab\\u{a}, Desai and Tait proved this conjecture for $n\\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $\\gamma=\\gamma(k)$ such that the same conclusion holds for $n\\ge \\Omega(k^{\\gamma(k)})$.\n  We prove a stronger uniform theorem for Nikiforov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06987","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/2606.06987/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2606.06987","created_at":"2026-06-08T01:04:39.713372+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.06987v1","created_at":"2026-06-08T01:04:39.713372+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.06987","created_at":"2026-06-08T01:04:39.713372+00:00"},{"alias_kind":"pith_short_12","alias_value":"EX7XG6KVLLR7","created_at":"2026-06-08T01:04:39.713372+00:00"},{"alias_kind":"pith_short_16","alias_value":"EX7XG6KVLLR7UT2T","created_at":"2026-06-08T01:04:39.713372+00:00"},{"alias_kind":"pith_short_8","alias_value":"EX7XG6KV","created_at":"2026-06-08T01:04:39.713372+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/EX7XG6KVLLR7UT2T7LN2A2VQC3","json":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3.json","graph_json":"https://pith.science/api/pith-number/EX7XG6KVLLR7UT2T7LN2A2VQC3/graph.json","events_json":"https://pith.science/api/pith-number/EX7XG6KVLLR7UT2T7LN2A2VQC3/events.json","paper":"https://pith.science/paper/EX7XG6KV"},"agent_actions":{"view_html":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3","download_json":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3.json","view_paper":"https://pith.science/paper/EX7XG6KV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.06987&json=true","fetch_graph":"https://pith.science/api/pith-number/EX7XG6KVLLR7UT2T7LN2A2VQC3/graph.json","fetch_events":"https://pith.science/api/pith-number/EX7XG6KVLLR7UT2T7LN2A2VQC3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3/action/storage_attestation","attest_author":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3/action/author_attestation","sign_citation":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3/action/citation_signature","submit_replication":"https://pith.science/pith/EX7XG6KVLLR7UT2T7LN2A2VQC3/action/replication_record"}},"created_at":"2026-06-08T01:04:39.713372+00:00","updated_at":"2026-06-08T01:04:39.713372+00:00"}