{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:7PFJPRZT4ZRQ4SZR2NY5UHSRFZ","short_pith_number":"pith:7PFJPRZT","canonical_record":{"source":{"id":"1904.03019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-04T02:56:48Z","cross_cats_sorted":[],"title_canon_sha256":"5e1f0d5d44d963fc1a855c37230fb0d7afc243277baf05db4680f12ba5a101e5","abstract_canon_sha256":"2566241b036cfc0b19bea92ababc802a7206610e273432e6bb11b01bb3398ffb"},"schema_version":"1.0"},"canonical_sha256":"fbca97c733e6630e4b31d371da1e512e53cd5d30f31771870f7730bc2be31f7a","source":{"kind":"arxiv","id":"1904.03019","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.03019","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"arxiv_version","alias_value":"1904.03019v1","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03019","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"pith_short_12","alias_value":"7PFJPRZT4ZRQ","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"7PFJPRZT4ZRQ4SZR","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"7PFJPRZT","created_at":"2026-05-18T12:33:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:7PFJPRZT4ZRQ4SZR2NY5UHSRFZ","target":"record","payload":{"canonical_record":{"source":{"id":"1904.03019","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-04T02:56:48Z","cross_cats_sorted":[],"title_canon_sha256":"5e1f0d5d44d963fc1a855c37230fb0d7afc243277baf05db4680f12ba5a101e5","abstract_canon_sha256":"2566241b036cfc0b19bea92ababc802a7206610e273432e6bb11b01bb3398ffb"},"schema_version":"1.0"},"canonical_sha256":"fbca97c733e6630e4b31d371da1e512e53cd5d30f31771870f7730bc2be31f7a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:18.865971Z","signature_b64":"QNdpKFVSksAxdXqn5E4a0s96qcnlbWLwNiaIa0hbsgwTqJtCUDF/fAqyjea15+L9+cuwwP3KU1NnwaTc8yCZAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fbca97c733e6630e4b31d371da1e512e53cd5d30f31771870f7730bc2be31f7a","last_reissued_at":"2026-05-17T23:49:18.865466Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:18.865466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.03019","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":"2026-05-17T23:49:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QQhCL88vSJoNRXOYpTwf5bBw3+w5a/nTGlU8eXO1HpFQ7+v3BXRiCl7IeZXxLi7GSbXCLtG4hLPcJhBLNzo0Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:05:43.424649Z"},"content_sha256":"580d6c377933080a7a386fc8c23fa93a30496fe8f2c54bd4ceed2075cca2a36e","schema_version":"1.0","event_id":"sha256:580d6c377933080a7a386fc8c23fa93a30496fe8f2c54bd4ceed2075cca2a36e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:7PFJPRZT4ZRQ4SZR2NY5UHSRFZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Guangjun Zhu, Hong Wang, Jiaqi Zhang, Li Xu","submitted_at":"2019-04-04T02:56:48Z","abstract_excerpt":"In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about \"rooted\" forest such that $w(x)\\geq 2$ if $d(x)\\neq 1$ cannot be dropped."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03019","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":""},"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":"2026-05-17T23:49:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FQB4Fr8TIRwGTWPzLBKi6GMhKXZlnzQE8mBw6ghy/ff4vIXFEiNBBG0cpO6tEcvWA9fce3P6ubxaCWidD6+BAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T23:05:43.425368Z"},"content_sha256":"7712f7b3f764f3531df566b78213911d37a940434eadd8b71f53dc6fd2e92a52","schema_version":"1.0","event_id":"sha256:7712f7b3f764f3531df566b78213911d37a940434eadd8b71f53dc6fd2e92a52"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/bundle.json","state_url":"https://pith.science/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/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":"2026-06-07T23:05:43Z","links":{"resolver":"https://pith.science/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ","bundle":"https://pith.science/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/bundle.json","state":"https://pith.science/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7PFJPRZT4ZRQ4SZR2NY5UHSRFZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:7PFJPRZT4ZRQ4SZR2NY5UHSRFZ","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":"2566241b036cfc0b19bea92ababc802a7206610e273432e6bb11b01bb3398ffb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-04T02:56:48Z","title_canon_sha256":"5e1f0d5d44d963fc1a855c37230fb0d7afc243277baf05db4680f12ba5a101e5"},"schema_version":"1.0","source":{"id":"1904.03019","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.03019","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"arxiv_version","alias_value":"1904.03019v1","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.03019","created_at":"2026-05-17T23:49:18Z"},{"alias_kind":"pith_short_12","alias_value":"7PFJPRZT4ZRQ","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_16","alias_value":"7PFJPRZT4ZRQ4SZR","created_at":"2026-05-18T12:33:12Z"},{"alias_kind":"pith_short_8","alias_value":"7PFJPRZT","created_at":"2026-05-18T12:33:12Z"}],"graph_snapshots":[{"event_id":"sha256:7712f7b3f764f3531df566b78213911d37a940434eadd8b71f53dc6fd2e92a52","target":"graph","created_at":"2026-05-17T23:49:18Z","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"},"paper":{"abstract_excerpt":"In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about \"rooted\" forest such that $w(x)\\geq 2$ if $d(x)\\neq 1$ cannot be dropped.","authors_text":"Guangjun Zhu, Hong Wang, Jiaqi Zhang, Li Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-04T02:56:48Z","title":"Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03019","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":"sha256:580d6c377933080a7a386fc8c23fa93a30496fe8f2c54bd4ceed2075cca2a36e","target":"record","created_at":"2026-05-17T23:49:18Z","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":"2566241b036cfc0b19bea92ababc802a7206610e273432e6bb11b01bb3398ffb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-04T02:56:48Z","title_canon_sha256":"5e1f0d5d44d963fc1a855c37230fb0d7afc243277baf05db4680f12ba5a101e5"},"schema_version":"1.0","source":{"id":"1904.03019","kind":"arxiv","version":1}},"canonical_sha256":"fbca97c733e6630e4b31d371da1e512e53cd5d30f31771870f7730bc2be31f7a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fbca97c733e6630e4b31d371da1e512e53cd5d30f31771870f7730bc2be31f7a","first_computed_at":"2026-05-17T23:49:18.865466Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:18.865466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QNdpKFVSksAxdXqn5E4a0s96qcnlbWLwNiaIa0hbsgwTqJtCUDF/fAqyjea15+L9+cuwwP3KU1NnwaTc8yCZAQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:18.865971Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.03019","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:580d6c377933080a7a386fc8c23fa93a30496fe8f2c54bd4ceed2075cca2a36e","sha256:7712f7b3f764f3531df566b78213911d37a940434eadd8b71f53dc6fd2e92a52"],"state_sha256":"9383130dbb3611c9ab8edfbf701f15f2f3ec47b9610f93c28b02b28bfb55c9d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LdB4rfa/GcsJm3Qs6YjUydvoirbhxEHWdnW7b6xj9mJH642LOEaKOmd2JdgcYcoMKvqi0A3++U5f8P6ks5YVDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T23:05:43.429018Z","bundle_sha256":"2c3704a99e6fddedd5d5e2da839856649a19468ec098792f9126e7a536b91d68"}}