{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:PCJEHRSH7WMWIXFCIWT54PDVEA","short_pith_number":"pith:PCJEHRSH","schema_version":"1.0","canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","source":{"kind":"arxiv","id":"2504.15772","version":2},"attestation_state":"computed","paper":{"title":"Laplacian eigenvalue distribution and girth of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dein Wong, Songnian Xu, Wenhao Zhen","submitted_at":"2025-04-22T10:28:41Z","abstract_excerpt":"Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\\leq n-g$. Moreover, we prove that $m_G(n-g+3,n]= n-g$ if and only if $G\\cong C_3$ or $G\\cong K_{3,2}$ or $G\\cong U_1$, where $U_1$ is obtained from a cycle by joining a single vertex with a vertex of this cycle."},"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":"2504.15772","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-22T10:28:41Z","cross_cats_sorted":[],"title_canon_sha256":"b8ddc7ebc2d59544d656bd9c1bbd6f119e78704a921a3389c7bd4e96e83c4f4b","abstract_canon_sha256":"085cf19c11bcc027b3d11e6dc3dfd0872d851c25acccbda967a770ef9c226163"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:25:09.878918Z","signature_b64":"qMEm8uJbelTP9a82zXjGe3oBmuOLqzgKRRttgot8294PrfEKA8L5640poMXhF83yxtwyR7OvsJ6ru4/n4VPrCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","last_reissued_at":"2026-07-05T11:25:09.878411Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:25:09.878411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Laplacian eigenvalue distribution and girth of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dein Wong, Songnian Xu, Wenhao Zhen","submitted_at":"2025-04-22T10:28:41Z","abstract_excerpt":"Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\\leq n-g$. Moreover, we prove that $m_G(n-g+3,n]= n-g$ if and only if $G\\cong C_3$ or $G\\cong K_{3,2}$ or $G\\cong U_1$, where $U_1$ is obtained from a cycle by joining a single vertex with a vertex of this cycle."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.15772","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.15772/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":"2504.15772","created_at":"2026-07-05T11:25:09.878468+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.15772v2","created_at":"2026-07-05T11:25:09.878468+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.15772","created_at":"2026-07-05T11:25:09.878468+00:00"},{"alias_kind":"pith_short_12","alias_value":"PCJEHRSH7WMW","created_at":"2026-07-05T11:25:09.878468+00:00"},{"alias_kind":"pith_short_16","alias_value":"PCJEHRSH7WMWIXFC","created_at":"2026-07-05T11:25:09.878468+00:00"},{"alias_kind":"pith_short_8","alias_value":"PCJEHRSH","created_at":"2026-07-05T11:25:09.878468+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.00921","citing_title":"Girth and Laplacian eigenvalue distribution","ref_index":33,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA","json":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA.json","graph_json":"https://pith.science/api/pith-number/PCJEHRSH7WMWIXFCIWT54PDVEA/graph.json","events_json":"https://pith.science/api/pith-number/PCJEHRSH7WMWIXFCIWT54PDVEA/events.json","paper":"https://pith.science/paper/PCJEHRSH"},"agent_actions":{"view_html":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA","download_json":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA.json","view_paper":"https://pith.science/paper/PCJEHRSH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.15772&json=true","fetch_graph":"https://pith.science/api/pith-number/PCJEHRSH7WMWIXFCIWT54PDVEA/graph.json","fetch_events":"https://pith.science/api/pith-number/PCJEHRSH7WMWIXFCIWT54PDVEA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/action/storage_attestation","attest_author":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/action/author_attestation","sign_citation":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/action/citation_signature","submit_replication":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/action/replication_record"}},"created_at":"2026-07-05T11:25:09.878468+00:00","updated_at":"2026-07-05T11:25:09.878468+00:00"}