{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:KEH5A75J6AAG2KRXR2DU6UIEJB","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":"443f97626ee0ebdba7a50af245d5dd9e6646a700d88d69415661838a598b755e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-01T09:24:24Z","title_canon_sha256":"31bbb4ad8fd8f8b0fd4b8f767d10b83c222b1eb7364b6b59b2f6086f0ccb030a"},"schema_version":"1.0","source":{"id":"2506.00921","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.00921","created_at":"2026-07-05T11:13:42Z"},{"alias_kind":"arxiv_version","alias_value":"2506.00921v1","created_at":"2026-07-05T11:13:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.00921","created_at":"2026-07-05T11:13:42Z"},{"alias_kind":"pith_short_12","alias_value":"KEH5A75J6AAG","created_at":"2026-07-05T11:13:42Z"},{"alias_kind":"pith_short_16","alias_value":"KEH5A75J6AAG2KRX","created_at":"2026-07-05T11:13:42Z"},{"alias_kind":"pith_short_8","alias_value":"KEH5A75J","created_at":"2026-07-05T11:13:42Z"}],"graph_snapshots":[{"event_id":"sha256:c6329e5f93ded23eee8cbc11e886950fbda5bc927492053c69afe8f613f7a612","target":"graph","created_at":"2026-07-05T11:13:42Z","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/2506.00921/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G$ be a connected graph of order $n$ with girth $g$. For $k=1,\\dots,\\min\\{g-1, n-g\\}$, let $n(G,k)$ be the number of Laplacian eigenvalues (counting multiplicities) of $G$ that fall inside the interval $[n-g-k+4,n]$. We prove that if $g\\ge 4$, then \\[ n(G,k)\\le n-g. \\] Those graphs achieving the bound for $k=1,2$ are determined. We also determine the graphs $G$ with $g=3$ such that $n(G,k)=n-1, n-2, n-3$.","authors_text":"Bo Zhou, Leyou Xu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-01T09:24:24Z","title":"Girth and Laplacian eigenvalue distribution"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.00921","kind":"arxiv","version":1},"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:bd069cf2a33847fef29dd61d23469c2d12e1526c648c193ddd0b0b7e93cd2b5c","target":"record","created_at":"2026-07-05T11:13:42Z","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":"443f97626ee0ebdba7a50af245d5dd9e6646a700d88d69415661838a598b755e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-06-01T09:24:24Z","title_canon_sha256":"31bbb4ad8fd8f8b0fd4b8f767d10b83c222b1eb7364b6b59b2f6086f0ccb030a"},"schema_version":"1.0","source":{"id":"2506.00921","kind":"arxiv","version":1}},"canonical_sha256":"510fd07fa9f0006d2a378e874f510448466f856b769c61c2d74c741e9686870f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"510fd07fa9f0006d2a378e874f510448466f856b769c61c2d74c741e9686870f","first_computed_at":"2026-07-05T11:13:42.857336Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:13:42.857336Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"33s2cAOO09WpSjJZqqrqdP6wkYwE8m8iLBR6Csai8FIniSjzXksVRQ6oaR32vOiKBg5MGvJyzf314852MK4HAw==","signature_status":"signed_v1","signed_at":"2026-07-05T11:13:42.857840Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.00921","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd069cf2a33847fef29dd61d23469c2d12e1526c648c193ddd0b0b7e93cd2b5c","sha256:c6329e5f93ded23eee8cbc11e886950fbda5bc927492053c69afe8f613f7a612"],"state_sha256":"9b66b2571a98f0a2a4d337701063dbe121caaa23541d73b93cc858c0f9ebf8cf"}