{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:A2JMCVWC2QQXHACTAJLQTBSYTS","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":"a4efe148162d71230e4d6d61959308063d21c1df33c00287dbdd8f4752655fb9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-01-03T13:08:22Z","title_canon_sha256":"24c1c2c83ec3bb8951e65439cb0a8211fbf840247d8512040adce2bd86a0ccb2"},"schema_version":"1.0","source":{"id":"1101.0515","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.0515","created_at":"2026-05-18T04:31:59Z"},{"alias_kind":"arxiv_version","alias_value":"1101.0515v2","created_at":"2026-05-18T04:31:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.0515","created_at":"2026-05-18T04:31:59Z"},{"alias_kind":"pith_short_12","alias_value":"A2JMCVWC2QQX","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"A2JMCVWC2QQXHACT","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"A2JMCVWC","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:17b393a4a07b653503f01577ab62519ca0ed05d9ea393fad9dab9a1bd1080e6a","target":"graph","created_at":"2026-05-18T04:31:59Z","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":"Let $G$ and $H$ be two simple graphs and let $G*H$ denotes the graph theoretical product of $G$ by $H$. In this paper we provide some results on graded Betti numbers, Castelnuovo-Mumford regularity, projective dimension, $h$-vector, and Hilbert series of $G*H$ in terms of that information of $G$ and $H$. To do this, we will provide explicit formulae to compute graded Betti numbers, $h$-vector, and Hilbert series of disjoint union of complexes. Also we will prove that the family of graphs whose regularity equal the maximum number of pairwise $3$-disjoint edges, is closed under product of graphs","authors_text":"Amir Mousivand","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-01-03T13:08:22Z","title":"Algebraic properties of product of graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.0515","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":"sha256:16f39a5bebc5e41493805352b21fc7155bf2e13668dea8cc023f2aef63b8daa2","target":"record","created_at":"2026-05-18T04:31:59Z","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":"a4efe148162d71230e4d6d61959308063d21c1df33c00287dbdd8f4752655fb9","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-01-03T13:08:22Z","title_canon_sha256":"24c1c2c83ec3bb8951e65439cb0a8211fbf840247d8512040adce2bd86a0ccb2"},"schema_version":"1.0","source":{"id":"1101.0515","kind":"arxiv","version":2}},"canonical_sha256":"0692c156c2d42173805302570986589c8486679fef9c9b58373b1c1118233c06","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0692c156c2d42173805302570986589c8486679fef9c9b58373b1c1118233c06","first_computed_at":"2026-05-18T04:31:59.772846Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:59.772846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Lw5YvYmm7pG9rYEbie+bft0vMmng/i8G89QJvEH8kByGktzSsJR5VN5AntEM0mRHCjpbh+iVHA+t+jwzLRtmDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:59.773369Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.0515","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16f39a5bebc5e41493805352b21fc7155bf2e13668dea8cc023f2aef63b8daa2","sha256:17b393a4a07b653503f01577ab62519ca0ed05d9ea393fad9dab9a1bd1080e6a"],"state_sha256":"da287546404a6a15f0458e3737f97a374682638293cf740de7c3521a94ef15ce"}