{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:FP3KL6SO2XE6RLQXEJLKYGHU2Q","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":"24cd70f692f7be2675733b64e7c248562bf3bd7dfd54b590e25f7d7cdb86aacf","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-05-16T13:51:35Z","title_canon_sha256":"00d3a24802a680e47c4246c39e3f7761dd823ff3de3c76d39d5223fe6e9770d7"},"schema_version":"1.0","source":{"id":"1205.3678","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.3678","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"arxiv_version","alias_value":"1205.3678v2","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.3678","created_at":"2026-05-18T03:32:45Z"},{"alias_kind":"pith_short_12","alias_value":"FP3KL6SO2XE6","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"FP3KL6SO2XE6RLQX","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"FP3KL6SO","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:a72aebf4981ade736a22cc50b166a4be38085b2b0c5fe424703f2801abc68c09","target":"graph","created_at":"2026-05-18T03:32:45Z","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":"We study weighted graphs and their \"edge ideals\" which are ideals in polynomial rings that are defined in terms of the graphs. We provide combinatorial descriptions of m-irreducible decompositions for the edge ideal of a weighted graph in terms of the combinatorics of \"weighted vertex covers\". We use these, for instance, to say when these ideals are m-unmixed. We explicitly describe which weighted cycles, suspensions, and trees are unmixed and which ones are Cohen-Macaulay, and we prove that all weighted complete graphs are Cohen-Macaulay.","authors_text":"Chelsey Paulsen, Sean Sather-Wagstaff","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-05-16T13:51:35Z","title":"Edge Ideals of Weighted Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.3678","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:f0e4bbcee375a45c967cda4d08b52bd3b2ddfd2c06d1606aef3dbacdac3c3220","target":"record","created_at":"2026-05-18T03:32:45Z","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":"24cd70f692f7be2675733b64e7c248562bf3bd7dfd54b590e25f7d7cdb86aacf","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-05-16T13:51:35Z","title_canon_sha256":"00d3a24802a680e47c4246c39e3f7761dd823ff3de3c76d39d5223fe6e9770d7"},"schema_version":"1.0","source":{"id":"1205.3678","kind":"arxiv","version":2}},"canonical_sha256":"2bf6a5fa4ed5c9e8ae172256ac18f4d41c5907a36d928afc52a75dc3002caa15","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2bf6a5fa4ed5c9e8ae172256ac18f4d41c5907a36d928afc52a75dc3002caa15","first_computed_at":"2026-05-18T03:32:45.238919Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:45.238919Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7JcQ5uu1/2jqGQjvh3AC0TOrqLXjf6yDXlh7XQstsSrubDEHJrEOM3YT9znLkk+Iy27uyyXYLNZqXaSmDboxDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:45.239740Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.3678","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f0e4bbcee375a45c967cda4d08b52bd3b2ddfd2c06d1606aef3dbacdac3c3220","sha256:a72aebf4981ade736a22cc50b166a4be38085b2b0c5fe424703f2801abc68c09"],"state_sha256":"8bf544deb55b827853daffe06687427dbebf6e0bb424cfe6e5f78450ccb6e38c"}