{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:H5HV6KMFVVALSFWTNFLIRV3LF4","short_pith_number":"pith:H5HV6KMF","canonical_record":{"source":{"id":"2506.08557","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T08:25:49Z","cross_cats_sorted":[],"title_canon_sha256":"6ebcd199d71bd95a9ebd92391a3c696c25a9140ac8335b10d95f2ce9e025c662","abstract_canon_sha256":"ccee416105c3dc240bfc6348cd4c92387cb9456b3d8ec26abc8b08e1003aa13d"},"schema_version":"1.0"},"canonical_sha256":"3f4f5f2985ad40b916d3695688d76b2f29e58ef540797ea80cc646fc50684fc2","source":{"kind":"arxiv","id":"2506.08557","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08557","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08557v1","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08557","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_12","alias_value":"H5HV6KMFVVAL","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_16","alias_value":"H5HV6KMFVVALSFWT","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_8","alias_value":"H5HV6KMF","created_at":"2026-07-05T11:19:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:H5HV6KMFVVALSFWTNFLIRV3LF4","target":"record","payload":{"canonical_record":{"source":{"id":"2506.08557","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T08:25:49Z","cross_cats_sorted":[],"title_canon_sha256":"6ebcd199d71bd95a9ebd92391a3c696c25a9140ac8335b10d95f2ce9e025c662","abstract_canon_sha256":"ccee416105c3dc240bfc6348cd4c92387cb9456b3d8ec26abc8b08e1003aa13d"},"schema_version":"1.0"},"canonical_sha256":"3f4f5f2985ad40b916d3695688d76b2f29e58ef540797ea80cc646fc50684fc2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:19:09.668530Z","signature_b64":"x4mmZchXchPmaloZDIt9xABqJk88XwAc5popVwofNVLYLqKIrerbl5DHRZ72fNHJb60rlpExvnZf774z6/7RDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f4f5f2985ad40b916d3695688d76b2f29e58ef540797ea80cc646fc50684fc2","last_reissued_at":"2026-07-05T11:19:09.668055Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:19:09.668055Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2506.08557","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T11:19:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OI4WyYedKfA05nVHSYNoQTa4VJECUhBeoaqCgERIjlurI8CWxVRu5LSscFuCWTaJiqLztcN+NlGRROYuX4BRBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:38:14.987344Z"},"content_sha256":"bca94c7d7f109d17406d290bf2e53d44645619f8459111e6ad9fb42cde53c7ee","schema_version":"1.0","event_id":"sha256:bca94c7d7f109d17406d290bf2e53d44645619f8459111e6ad9fb42cde53c7ee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:H5HV6KMFVVALSFWTNFLIRV3LF4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the maximal matchings of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kai Deng, Lingjuan Shi, Wei Li","submitted_at":"2025-06-10T08:25:49Z","abstract_excerpt":"An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\\Psi(G)$. In this paper, an algorithm to count $\\Psi(T)$ for a tree $T$ is given. We show that for any tree $T$ with $n$ vertices, $\\Psi(T)\\geq\\lceil\\frac{n}{2}\\rceil$, and the tree which obtained the lower bound is characterized."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08557","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/2506.08557/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":"2026-07-05T11:19:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"c8KhfJ0RL4a7DGAtxdisf0BgggAKUzvmmu+P8pSjt1UQ3/gZHUpBb4RgUvqo9un+G4WRFRmhqnZSxxwpzP8WCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:38:14.987833Z"},"content_sha256":"33ae5280b402a7e5adf72891ed68cfc459cb83871377c6f513b588d306c3a124","schema_version":"1.0","event_id":"sha256:33ae5280b402a7e5adf72891ed68cfc459cb83871377c6f513b588d306c3a124"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/bundle.json","state_url":"https://pith.science/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/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":"2026-08-08T10:38:14Z","links":{"resolver":"https://pith.science/pith/H5HV6KMFVVALSFWTNFLIRV3LF4","bundle":"https://pith.science/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/bundle.json","state":"https://pith.science/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/H5HV6KMFVVALSFWTNFLIRV3LF4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:H5HV6KMFVVALSFWTNFLIRV3LF4","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":"ccee416105c3dc240bfc6348cd4c92387cb9456b3d8ec26abc8b08e1003aa13d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T08:25:49Z","title_canon_sha256":"6ebcd199d71bd95a9ebd92391a3c696c25a9140ac8335b10d95f2ce9e025c662"},"schema_version":"1.0","source":{"id":"2506.08557","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2506.08557","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"arxiv_version","alias_value":"2506.08557v1","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08557","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_12","alias_value":"H5HV6KMFVVAL","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_16","alias_value":"H5HV6KMFVVALSFWT","created_at":"2026-07-05T11:19:09Z"},{"alias_kind":"pith_short_8","alias_value":"H5HV6KMF","created_at":"2026-07-05T11:19:09Z"}],"graph_snapshots":[{"event_id":"sha256:33ae5280b402a7e5adf72891ed68cfc459cb83871377c6f513b588d306c3a124","target":"graph","created_at":"2026-07-05T11:19:09Z","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/2506.08557/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\\Psi(G)$. In this paper, an algorithm to count $\\Psi(T)$ for a tree $T$ is given. We show that for any tree $T$ with $n$ vertices, $\\Psi(T)\\geq\\lceil\\frac{n}{2}\\rceil$, and the tree which obtained the lower bound is characterized.","authors_text":"Kai Deng, Lingjuan Shi, Wei Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T08:25:49Z","title":"On the maximal matchings of trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08557","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:bca94c7d7f109d17406d290bf2e53d44645619f8459111e6ad9fb42cde53c7ee","target":"record","created_at":"2026-07-05T11:19:09Z","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":"ccee416105c3dc240bfc6348cd4c92387cb9456b3d8ec26abc8b08e1003aa13d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-06-10T08:25:49Z","title_canon_sha256":"6ebcd199d71bd95a9ebd92391a3c696c25a9140ac8335b10d95f2ce9e025c662"},"schema_version":"1.0","source":{"id":"2506.08557","kind":"arxiv","version":1}},"canonical_sha256":"3f4f5f2985ad40b916d3695688d76b2f29e58ef540797ea80cc646fc50684fc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3f4f5f2985ad40b916d3695688d76b2f29e58ef540797ea80cc646fc50684fc2","first_computed_at":"2026-07-05T11:19:09.668055Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T11:19:09.668055Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"x4mmZchXchPmaloZDIt9xABqJk88XwAc5popVwofNVLYLqKIrerbl5DHRZ72fNHJb60rlpExvnZf774z6/7RDA==","signature_status":"signed_v1","signed_at":"2026-07-05T11:19:09.668530Z","signed_message":"canonical_sha256_bytes"},"source_id":"2506.08557","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bca94c7d7f109d17406d290bf2e53d44645619f8459111e6ad9fb42cde53c7ee","sha256:33ae5280b402a7e5adf72891ed68cfc459cb83871377c6f513b588d306c3a124"],"state_sha256":"5ca1eaf85926a213f592032098d3424d6cce3cb5f1748b603ccb913e3a3119fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sQtlaDqLZ9/ca7K/W8IF3Oiin/8o/VksmXBxKyWgyPzj2wQsV0LoIoHHOd++JsDwfLc9VYNyqhE+ZPZ29o+xDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T10:38:14.993280Z","bundle_sha256":"d413baff6b353b8720cd2289f6e7d4440e5494b60a4cf8c69f1d9e386fddd3a8"}}