{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PCJEHRSH7WMWIXFCIWT54PDVEA","short_pith_number":"pith:PCJEHRSH","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"},"canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","source":{"kind":"arxiv","id":"2504.15772","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.15772","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"arxiv_version","alias_value":"2504.15772v2","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.15772","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_12","alias_value":"PCJEHRSH7WMW","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_16","alias_value":"PCJEHRSH7WMWIXFC","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_8","alias_value":"PCJEHRSH","created_at":"2026-07-05T11:25:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PCJEHRSH7WMWIXFCIWT54PDVEA","target":"record","payload":{"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"},"canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","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"},"source_kind":"arxiv","source_id":"2504.15772","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:25:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"f44/n4ANl72n01yRe+nsn93C47w3C71fVou/O+XY0IWg/t0mS1PjypL74zcveKrl8QoWYPwTKItWLIWliMypBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T08:27:03.141414Z"},"content_sha256":"b965d2fc71506cfe2e2f6a5825ce010bb6b111345ae6cea0935bec74343a6676","schema_version":"1.0","event_id":"sha256:b965d2fc71506cfe2e2f6a5825ce010bb6b111345ae6cea0935bec74343a6676"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PCJEHRSH7WMWIXFCIWT54PDVEA","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:25:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YGY+uWhGIW1obY9eN0/t+iWTuOJAj9gTVF8HRuYi7qq5v6ek70Y2YtH1Bgpot379/dVoDphyl0QeTZeUiTgtAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T08:27:03.141958Z"},"content_sha256":"96b4359bcbc252a4cb19883d28ea6824d22f02d077cca644c37d4caf4335258e","schema_version":"1.0","event_id":"sha256:96b4359bcbc252a4cb19883d28ea6824d22f02d077cca644c37d4caf4335258e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/bundle.json","state_url":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-08T08:27:03Z","links":{"resolver":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA","bundle":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/bundle.json","state":"https://pith.science/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PCJEHRSH7WMWIXFCIWT54PDVEA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PCJEHRSH7WMWIXFCIWT54PDVEA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"085cf19c11bcc027b3d11e6dc3dfd0872d851c25acccbda967a770ef9c226163","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-22T10:28:41Z","title_canon_sha256":"b8ddc7ebc2d59544d656bd9c1bbd6f119e78704a921a3389c7bd4e96e83c4f4b"},"schema_version":"1.0","source":{"id":"2504.15772","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.15772","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"arxiv_version","alias_value":"2504.15772v2","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.15772","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_12","alias_value":"PCJEHRSH7WMW","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_16","alias_value":"PCJEHRSH7WMWIXFC","created_at":"2026-07-05T11:25:09Z"},{"alias_kind":"pith_short_8","alias_value":"PCJEHRSH","created_at":"2026-07-05T11:25:09Z"}],"graph_snapshots":[{"event_id":"sha256:96b4359bcbc252a4cb19883d28ea6824d22f02d077cca644c37d4caf4335258e","target":"graph","created_at":"2026-07-05T11:25:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2504.15772/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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.","authors_text":"Dein Wong, Songnian Xu, Wenhao Zhen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-22T10:28:41Z","title":"Laplacian eigenvalue distribution and girth of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.15772","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b965d2fc71506cfe2e2f6a5825ce010bb6b111345ae6cea0935bec74343a6676","target":"record","created_at":"2026-07-05T11:25:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"085cf19c11bcc027b3d11e6dc3dfd0872d851c25acccbda967a770ef9c226163","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-22T10:28:41Z","title_canon_sha256":"b8ddc7ebc2d59544d656bd9c1bbd6f119e78704a921a3389c7bd4e96e83c4f4b"},"schema_version":"1.0","source":{"id":"2504.15772","kind":"arxiv","version":2}},"canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"789243c647fd99645ca245a7de3c75200ba57c1db89483ce73f5d8f29780a051","first_computed_at":"2026-07-05T11:25:09.878411Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:25:09.878411Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qMEm8uJbelTP9a82zXjGe3oBmuOLqzgKRRttgot8294PrfEKA8L5640poMXhF83yxtwyR7OvsJ6ru4/n4VPrCw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:25:09.878918Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.15772","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b965d2fc71506cfe2e2f6a5825ce010bb6b111345ae6cea0935bec74343a6676","sha256:96b4359bcbc252a4cb19883d28ea6824d22f02d077cca644c37d4caf4335258e"],"state_sha256":"9441a3e897d3cf7380e3c6e239497ae2e14d7f6ffdf08be708212f2e7e62099c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4hh2sJEkYTcgJh4zc7lisgZ2Ncs38L7QPP4yZpW/shoJYZNzf87/1AQkxOf+Y0MX5/QCQeHnZ4NQDg8yPE81Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T08:27:03.145592Z","bundle_sha256":"5229a72ffca9286c0f6683793062c9999a8f1344f2fd5fcdb5b8574c7fb1a844"}}