{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:WHS7E5EUB4M7LOZSGCU5RFEEEB","short_pith_number":"pith:WHS7E5EU","canonical_record":{"source":{"id":"1907.02743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-05T09:38:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c92f9ff73d41fb24f303a4a3d401d3bc96ecddee7dcd63480127d907fa75d9fc","abstract_canon_sha256":"fb6d9a1977b4f32a3a84a0d6c86e34a6ab09f3b5084d5cd36a9e8100c1ecb641"},"schema_version":"1.0"},"canonical_sha256":"b1e5f274940f19f5bb3230a9d89484206c5dddb0381a74c2c786529dc246ec6e","source":{"kind":"arxiv","id":"1907.02743","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.02743","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"arxiv_version","alias_value":"1907.02743v1","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.02743","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"pith_short_12","alias_value":"WHS7E5EUB4M7","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WHS7E5EUB4M7LOZS","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WHS7E5EU","created_at":"2026-05-18T12:33:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:WHS7E5EUB4M7LOZSGCU5RFEEEB","target":"record","payload":{"canonical_record":{"source":{"id":"1907.02743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-05T09:38:21Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"c92f9ff73d41fb24f303a4a3d401d3bc96ecddee7dcd63480127d907fa75d9fc","abstract_canon_sha256":"fb6d9a1977b4f32a3a84a0d6c86e34a6ab09f3b5084d5cd36a9e8100c1ecb641"},"schema_version":"1.0"},"canonical_sha256":"b1e5f274940f19f5bb3230a9d89484206c5dddb0381a74c2c786529dc246ec6e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:23.319685Z","signature_b64":"Rkwyr86Eg3KBAlL9bF0lPPTBCM4+v8HXMsoz7BbUWPBBSA05t8ChTQfSDI7sTrRxIQdqw+ujTi27DsQbQbruDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1e5f274940f19f5bb3230a9d89484206c5dddb0381a74c2c786529dc246ec6e","last_reissued_at":"2026-05-17T23:41:23.318847Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:23.318847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1907.02743","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":"2026-05-17T23:41:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"btKvHdhVG60b4Y5cw48PH4Tcvep9S4KaVXF3Tg5+w+bb6VwMDnH9KsopFV4oa/o9q/1f47yR1OP94q0XsAPSCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:52:14.927106Z"},"content_sha256":"38be3be7934374c593b5131dcb5c5e80eb904f354f4340a2bfd76f96e0a0c5df","schema_version":"1.0","event_id":"sha256:38be3be7934374c593b5131dcb5c5e80eb904f354f4340a2bfd76f96e0a0c5df"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:WHS7E5EUB4M7LOZSGCU5RFEEEB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity of symbolic powers of edge ideals of Cameron-Walker graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"S. A. Seyed Fakhari","submitted_at":"2019-07-05T09:38:21Z","abstract_excerpt":"A Cameron-Walker graph is a graph for which the matching number and the induced matching number are the same. Assume that $G$ is a Cameron-Walker graph with edge ideal $I(G)$, and let $\\ind-match(G)$ be the induced matching number of $G$. It is shown that for every integer $s\\geq 1$, we have the equality ${\\rm reg}(I(G)^{(s)})=2s+\\ind-match(G)-1$, where $I(G)^{(s)}$ denotes the $s$-th symbolic power of $I(G)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02743","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":"2026-05-17T23:41:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tyyBHXQ74uHqiPfdUACYXCePWd7eRA8NqBmkapc0qr3eJYWb+fCpQCucQ5YAa1dLvyjafr3tiXj2vhHRyFpqBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T19:52:14.927815Z"},"content_sha256":"276ddc60b553dfb2d41df2f13636735fabc094e5e82ace755dfbcce30de0e51e","schema_version":"1.0","event_id":"sha256:276ddc60b553dfb2d41df2f13636735fabc094e5e82ace755dfbcce30de0e51e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/bundle.json","state_url":"https://pith.science/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/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":"2026-05-25T19:52:14Z","links":{"resolver":"https://pith.science/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB","bundle":"https://pith.science/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/bundle.json","state":"https://pith.science/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WHS7E5EUB4M7LOZSGCU5RFEEEB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:WHS7E5EUB4M7LOZSGCU5RFEEEB","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":"fb6d9a1977b4f32a3a84a0d6c86e34a6ab09f3b5084d5cd36a9e8100c1ecb641","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-05T09:38:21Z","title_canon_sha256":"c92f9ff73d41fb24f303a4a3d401d3bc96ecddee7dcd63480127d907fa75d9fc"},"schema_version":"1.0","source":{"id":"1907.02743","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1907.02743","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"arxiv_version","alias_value":"1907.02743v1","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1907.02743","created_at":"2026-05-17T23:41:23Z"},{"alias_kind":"pith_short_12","alias_value":"WHS7E5EUB4M7","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"WHS7E5EUB4M7LOZS","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"WHS7E5EU","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:276ddc60b553dfb2d41df2f13636735fabc094e5e82ace755dfbcce30de0e51e","target":"graph","created_at":"2026-05-17T23:41:23Z","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":"A Cameron-Walker graph is a graph for which the matching number and the induced matching number are the same. Assume that $G$ is a Cameron-Walker graph with edge ideal $I(G)$, and let $\\ind-match(G)$ be the induced matching number of $G$. It is shown that for every integer $s\\geq 1$, we have the equality ${\\rm reg}(I(G)^{(s)})=2s+\\ind-match(G)-1$, where $I(G)^{(s)}$ denotes the $s$-th symbolic power of $I(G)$.","authors_text":"S. A. Seyed Fakhari","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-05T09:38:21Z","title":"Regularity of symbolic powers of edge ideals of Cameron-Walker graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02743","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":"sha256:38be3be7934374c593b5131dcb5c5e80eb904f354f4340a2bfd76f96e0a0c5df","target":"record","created_at":"2026-05-17T23:41:23Z","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":"fb6d9a1977b4f32a3a84a0d6c86e34a6ab09f3b5084d5cd36a9e8100c1ecb641","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2019-07-05T09:38:21Z","title_canon_sha256":"c92f9ff73d41fb24f303a4a3d401d3bc96ecddee7dcd63480127d907fa75d9fc"},"schema_version":"1.0","source":{"id":"1907.02743","kind":"arxiv","version":1}},"canonical_sha256":"b1e5f274940f19f5bb3230a9d89484206c5dddb0381a74c2c786529dc246ec6e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b1e5f274940f19f5bb3230a9d89484206c5dddb0381a74c2c786529dc246ec6e","first_computed_at":"2026-05-17T23:41:23.318847Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:23.318847Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rkwyr86Eg3KBAlL9bF0lPPTBCM4+v8HXMsoz7BbUWPBBSA05t8ChTQfSDI7sTrRxIQdqw+ujTi27DsQbQbruDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:23.319685Z","signed_message":"canonical_sha256_bytes"},"source_id":"1907.02743","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:38be3be7934374c593b5131dcb5c5e80eb904f354f4340a2bfd76f96e0a0c5df","sha256:276ddc60b553dfb2d41df2f13636735fabc094e5e82ace755dfbcce30de0e51e"],"state_sha256":"8782f9db1894eb58c2aad09cf007ad4cc8dd4f5194d43d0dcc3d49beef0b3da5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JVTpj02bnTMkM1ps7szmGlCA2R+Q92u3Xh9gIHRp82LVBqgKf6b8/2Y+WlbTUlvtbZqnFKyRcEqtmoU+ZCf8Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T19:52:14.931612Z","bundle_sha256":"f3284d85a6f21a71abcdf643dad770d9bbd25fdf6fa990e856347fbab4ed94af"}}