{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:5A5YHB2EDI72B2QFIBA4JV62PW","short_pith_number":"pith:5A5YHB2E","canonical_record":{"source":{"id":"2303.11503","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-20T23:47:36Z","cross_cats_sorted":[],"title_canon_sha256":"a5b29d3d440dfbde45eeeaa873031cedeeaba2daf8cbe6d9cb5257f7fee4db84","abstract_canon_sha256":"ac8c2de66adf5832d5a29f4009df63a03e85945c6c59be4d9a72f94cb487eb4d"},"schema_version":"1.0"},"canonical_sha256":"e83b8387441a3fa0ea054041c4d7da7db56ab8781d6fa35b6d0ccd5d00a848b8","source":{"kind":"arxiv","id":"2303.11503","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.11503","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"2303.11503v2","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.11503","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"5A5YHB2EDI72","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_16","alias_value":"5A5YHB2EDI72B2QF","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_8","alias_value":"5A5YHB2E","created_at":"2026-07-05T06:22:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:5A5YHB2EDI72B2QFIBA4JV62PW","target":"record","payload":{"canonical_record":{"source":{"id":"2303.11503","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-20T23:47:36Z","cross_cats_sorted":[],"title_canon_sha256":"a5b29d3d440dfbde45eeeaa873031cedeeaba2daf8cbe6d9cb5257f7fee4db84","abstract_canon_sha256":"ac8c2de66adf5832d5a29f4009df63a03e85945c6c59be4d9a72f94cb487eb4d"},"schema_version":"1.0"},"canonical_sha256":"e83b8387441a3fa0ea054041c4d7da7db56ab8781d6fa35b6d0ccd5d00a848b8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:22:01.726647Z","signature_b64":"jtm1o+RuzouTdpMs4eBxHUiFZGDsRrqQ3bIPHSv6kUjpAoYv9D6kU4hPNQnPRvsN5ttr0lShtLkYv3xhDNrgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e83b8387441a3fa0ea054041c4d7da7db56ab8781d6fa35b6d0ccd5d00a848b8","last_reissued_at":"2026-07-05T06:22:01.726164Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:22:01.726164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.11503","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":"2026-07-05T06:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iA+b8Bx/p9c4R5+aN4w3aBdu2PI+PiRdOfh8w5dyptJWw3CzMBNYGj+T8a99D8VDL6q7UCfQfLCr/2NpDVBbDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:11:13.571057Z"},"content_sha256":"210265e9458b30def843aa1bc996e51c6d9bc66e52d40e554f304ad74a7a764d","schema_version":"1.0","event_id":"sha256:210265e9458b30def843aa1bc996e51c6d9bc66e52d40e554f304ad74a7a764d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:5A5YHB2EDI72B2QFIBA4JV62PW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Zhou, Leyou Xu","submitted_at":"2023-03-20T23:47:36Z","abstract_excerpt":"Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\\ge 2$ that is not a path, the number of Laplacian eigenvalues in the interval $[n-d+2,n]$ is at most $n-d$. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.11503","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/2303.11503/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":"2026-07-05T06:22:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B2067Y207/5PPeSGb/btjJu2DqW/fTuyk6FImhe+RLdzD1nmeSu3Zb6zoTDhQGS93sT0tgUUozYYTaSZqT3CDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-03T20:11:13.571547Z"},"content_sha256":"db8ac3f4e154c0d4ba4e24e0a78979b036af417cbb9d1dd15417a51fce879012","schema_version":"1.0","event_id":"sha256:db8ac3f4e154c0d4ba4e24e0a78979b036af417cbb9d1dd15417a51fce879012"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5A5YHB2EDI72B2QFIBA4JV62PW/bundle.json","state_url":"https://pith.science/pith/5A5YHB2EDI72B2QFIBA4JV62PW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5A5YHB2EDI72B2QFIBA4JV62PW/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":"2026-08-03T20:11:13Z","links":{"resolver":"https://pith.science/pith/5A5YHB2EDI72B2QFIBA4JV62PW","bundle":"https://pith.science/pith/5A5YHB2EDI72B2QFIBA4JV62PW/bundle.json","state":"https://pith.science/pith/5A5YHB2EDI72B2QFIBA4JV62PW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5A5YHB2EDI72B2QFIBA4JV62PW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:5A5YHB2EDI72B2QFIBA4JV62PW","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":"ac8c2de66adf5832d5a29f4009df63a03e85945c6c59be4d9a72f94cb487eb4d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-20T23:47:36Z","title_canon_sha256":"a5b29d3d440dfbde45eeeaa873031cedeeaba2daf8cbe6d9cb5257f7fee4db84"},"schema_version":"1.0","source":{"id":"2303.11503","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.11503","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"arxiv_version","alias_value":"2303.11503v2","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.11503","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_12","alias_value":"5A5YHB2EDI72","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_16","alias_value":"5A5YHB2EDI72B2QF","created_at":"2026-07-05T06:22:01Z"},{"alias_kind":"pith_short_8","alias_value":"5A5YHB2E","created_at":"2026-07-05T06:22:01Z"}],"graph_snapshots":[{"event_id":"sha256:db8ac3f4e154c0d4ba4e24e0a78979b036af417cbb9d1dd15417a51fce879012","target":"graph","created_at":"2026-07-05T06:22:01Z","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/2303.11503/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\\ge 2$ that is not a path, the number of Laplacian eigenvalues in the interval $[n-d+2,n]$ is at most $n-d$. We show that the conjecture is true, and give a complete characterization of graphs for which the conjectured bound is attained. This establishes an interesting relation between the spectral and classical parameters.","authors_text":"Bo Zhou, Leyou Xu","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-20T23:47:36Z","title":"Proof of a conjecture on distribution of Laplacian eigenvalues and diameter, and beyond"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.11503","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":"sha256:210265e9458b30def843aa1bc996e51c6d9bc66e52d40e554f304ad74a7a764d","target":"record","created_at":"2026-07-05T06:22:01Z","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":"ac8c2de66adf5832d5a29f4009df63a03e85945c6c59be4d9a72f94cb487eb4d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-03-20T23:47:36Z","title_canon_sha256":"a5b29d3d440dfbde45eeeaa873031cedeeaba2daf8cbe6d9cb5257f7fee4db84"},"schema_version":"1.0","source":{"id":"2303.11503","kind":"arxiv","version":2}},"canonical_sha256":"e83b8387441a3fa0ea054041c4d7da7db56ab8781d6fa35b6d0ccd5d00a848b8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e83b8387441a3fa0ea054041c4d7da7db56ab8781d6fa35b6d0ccd5d00a848b8","first_computed_at":"2026-07-05T06:22:01.726164Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:22:01.726164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jtm1o+RuzouTdpMs4eBxHUiFZGDsRrqQ3bIPHSv6kUjpAoYv9D6kU4hPNQnPRvsN5ttr0lShtLkYv3xhDNrgCw==","signature_status":"signed_v1","signed_at":"2026-07-05T06:22:01.726647Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.11503","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:210265e9458b30def843aa1bc996e51c6d9bc66e52d40e554f304ad74a7a764d","sha256:db8ac3f4e154c0d4ba4e24e0a78979b036af417cbb9d1dd15417a51fce879012"],"state_sha256":"d575b91395bf720adfd2642b96fbb3bd6b37dde1d94323dfd54e67b90ae7f1f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qUgrdv+JmScUBX4qQWCCn45oBkiAdu6IbGsaj6bNqMSjrebKdua1zyB5C4fzloDHu6m7fraG6qcROPmbvgtNBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-03T20:11:13.575046Z","bundle_sha256":"0b878ea73149d6ae508e0b350b80821191bc7eb6012c1d99975f19e47ec7362f"}}