{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:5P6WFISW4G46X2AIQ4R7LJ2D5V","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":"7af49d1b427e255b63145805e1dbefb734c3b66c0aeeadd949919aecb1e2e546","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-08T15:20:40Z","title_canon_sha256":"1d25346fc35f0f7d41972ea279e8e96d255beb1982b14c6bcd05b7edc1078b6b"},"schema_version":"1.0","source":{"id":"1904.04682","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.04682","created_at":"2026-05-17T23:44:28Z"},{"alias_kind":"arxiv_version","alias_value":"1904.04682v1","created_at":"2026-05-17T23:44:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.04682","created_at":"2026-05-17T23:44:28Z"},{"alias_kind":"pith_short_12","alias_value":"5P6WFISW4G46","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"5P6WFISW4G46X2AI","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"5P6WFISW","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:079549b38bf81727edcbc24ddbb9ea5aa65929674054409361e0e6c51907a0b5","target":"graph","created_at":"2026-05-17T23:44:28Z","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 the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented $m$-partite graphs. These formulas are functions of the weight and number of vertices. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.","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":"2019-04-08T15:20:40Z","title":"Projective dimension and regularity of edge ideals of some vertex-weighted oriented $m$-partite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.04682","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":"sha256:2b8108b07dbd135dc2811144307a36081911efceb79885bb64373c52814870d9","target":"record","created_at":"2026-05-17T23:44:28Z","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":"7af49d1b427e255b63145805e1dbefb734c3b66c0aeeadd949919aecb1e2e546","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-04-08T15:20:40Z","title_canon_sha256":"1d25346fc35f0f7d41972ea279e8e96d255beb1982b14c6bcd05b7edc1078b6b"},"schema_version":"1.0","source":{"id":"1904.04682","kind":"arxiv","version":1}},"canonical_sha256":"ebfd62a256e1b9ebe8088723f5a743ed66feaa34bb1ec6d4bc7dfdab3bc6ba43","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ebfd62a256e1b9ebe8088723f5a743ed66feaa34bb1ec6d4bc7dfdab3bc6ba43","first_computed_at":"2026-05-17T23:44:28.779329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:44:28.779329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"N3RURjMOtUPk+0r0eL/F/zZu6mxlzScgmT5GwTI5JxBb+rwY4XJ4fHPGYQ1/4GTVD8dvfHMgs31lfFkJAPPRCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:44:28.780007Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.04682","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b8108b07dbd135dc2811144307a36081911efceb79885bb64373c52814870d9","sha256:079549b38bf81727edcbc24ddbb9ea5aa65929674054409361e0e6c51907a0b5"],"state_sha256":"81fe8e9633e0a0bb882afa3ae40520e240e3a96263a2d2cb6d8cdd76323fe3cf"}