{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:7UVWZJBBFGMCHO3T2IY4BEGZ3D","short_pith_number":"pith:7UVWZJBB","schema_version":"1.0","canonical_sha256":"fd2b6ca421299823bb73d231c090d9d8dabe7656520a6ea0fb8d9f693e8db639","source":{"kind":"arxiv","id":"2508.05678","version":1},"attestation_state":"computed","paper":{"title":"Spectral conditions for graphs to contain $k$-factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wenqian Zhang, Xinying Tang","submitted_at":"2025-08-05T12:54:36Z","abstract_excerpt":"Let $G$ be a graph. The spectral radius $\\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\\geq1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers satisfying $k\\geq2,kn\\equiv0~(\\mod2)$ and $n\\geq\\max\\left\\{k^{2}+6k+7,20k+10\\right\\}$. Let $G$ be a graph of order $n$ and with minimum degree at least $k$. In this paper, we give a sharp lower bound of $\\rho(G)$ to guarantee that $G$ contains a $k$-factor."},"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":"2508.05678","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-08-05T12:54:36Z","cross_cats_sorted":[],"title_canon_sha256":"a07924d735dad684377e67f203fa5a3537e42961d91469a1ca88acdb1af0c9fe","abstract_canon_sha256":"949fe734c1744d4846c6cd73a5a86993344f0220bc2dfba391e76f3d8c174076"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:50:41.411591Z","signature_b64":"2m5/37/etJ4SbhRH01gK9hqJ3Jw9QX+jeIjBjW/gCHgwxTSjTMvkZc0XTUhet1vEo62DQGKN7UbhPsKFBGP4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd2b6ca421299823bb73d231c090d9d8dabe7656520a6ea0fb8d9f693e8db639","last_reissued_at":"2026-07-05T11:50:41.411089Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:50:41.411089Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spectral conditions for graphs to contain $k$-factors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wenqian Zhang, Xinying Tang","submitted_at":"2025-08-05T12:54:36Z","abstract_excerpt":"Let $G$ be a graph. The spectral radius $\\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\\geq1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers satisfying $k\\geq2,kn\\equiv0~(\\mod2)$ and $n\\geq\\max\\left\\{k^{2}+6k+7,20k+10\\right\\}$. Let $G$ be a graph of order $n$ and with minimum degree at least $k$. In this paper, we give a sharp lower bound of $\\rho(G)$ to guarantee that $G$ contains a $k$-factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.05678","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/2508.05678/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":"2508.05678","created_at":"2026-07-05T11:50:41.411148+00:00"},{"alias_kind":"arxiv_version","alias_value":"2508.05678v1","created_at":"2026-07-05T11:50:41.411148+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2508.05678","created_at":"2026-07-05T11:50:41.411148+00:00"},{"alias_kind":"pith_short_12","alias_value":"7UVWZJBBFGMC","created_at":"2026-07-05T11:50:41.411148+00:00"},{"alias_kind":"pith_short_16","alias_value":"7UVWZJBBFGMCHO3T","created_at":"2026-07-05T11:50:41.411148+00:00"},{"alias_kind":"pith_short_8","alias_value":"7UVWZJBB","created_at":"2026-07-05T11:50:41.411148+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.00769","citing_title":"A note on the spectral radius and $[a,b]$-factor of graphs","ref_index":26,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D","json":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D.json","graph_json":"https://pith.science/api/pith-number/7UVWZJBBFGMCHO3T2IY4BEGZ3D/graph.json","events_json":"https://pith.science/api/pith-number/7UVWZJBBFGMCHO3T2IY4BEGZ3D/events.json","paper":"https://pith.science/paper/7UVWZJBB"},"agent_actions":{"view_html":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D","download_json":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D.json","view_paper":"https://pith.science/paper/7UVWZJBB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2508.05678&json=true","fetch_graph":"https://pith.science/api/pith-number/7UVWZJBBFGMCHO3T2IY4BEGZ3D/graph.json","fetch_events":"https://pith.science/api/pith-number/7UVWZJBBFGMCHO3T2IY4BEGZ3D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D/action/storage_attestation","attest_author":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D/action/author_attestation","sign_citation":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D/action/citation_signature","submit_replication":"https://pith.science/pith/7UVWZJBBFGMCHO3T2IY4BEGZ3D/action/replication_record"}},"created_at":"2026-07-05T11:50:41.411148+00:00","updated_at":"2026-07-05T11:50:41.411148+00:00"}