{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:UDURJ7LHV3I5OFQJPE5YB346JB","short_pith_number":"pith:UDURJ7LH","canonical_record":{"source":{"id":"1112.5157","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T20:58:37Z","cross_cats_sorted":[],"title_canon_sha256":"c58da8b5393739f5a1a841945cd13e46707bb31c9b661b2f08dd8256585e5a29","abstract_canon_sha256":"a2cebc5352d192e46a8ee4e28a47e3121ef0b811b03e8bb06846ce2763a00210"},"schema_version":"1.0"},"canonical_sha256":"a0e914fd67aed1d71609793b80ef9e48432fd1700e5b3c648caad00552f96acf","source":{"kind":"arxiv","id":"1112.5157","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5157","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5157v1","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5157","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"UDURJ7LHV3I5","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UDURJ7LHV3I5OFQJ","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UDURJ7LH","created_at":"2026-05-18T12:26:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:UDURJ7LHV3I5OFQJPE5YB346JB","target":"record","payload":{"canonical_record":{"source":{"id":"1112.5157","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T20:58:37Z","cross_cats_sorted":[],"title_canon_sha256":"c58da8b5393739f5a1a841945cd13e46707bb31c9b661b2f08dd8256585e5a29","abstract_canon_sha256":"a2cebc5352d192e46a8ee4e28a47e3121ef0b811b03e8bb06846ce2763a00210"},"schema_version":"1.0"},"canonical_sha256":"a0e914fd67aed1d71609793b80ef9e48432fd1700e5b3c648caad00552f96acf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:48.575003Z","signature_b64":"8X5sJZ3o6bsmj77Yc/PUWAtGrPaGZm/j1LnA1Y3FWUiyInCew/oBvOo8kXNvbpWldq/xSn+73yLrOlup/NfaAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0e914fd67aed1d71609793b80ef9e48432fd1700e5b3c648caad00552f96acf","last_reissued_at":"2026-05-18T04:05:48.574214Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:48.574214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.5157","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":"2026-05-18T04:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2/KdH3OK9HvI+0StuZvOyJQnGyNtPhQnKAYOmPZnWIDBexDrc0qgf4MifXPIfKA22XzLRgr8j5z2WJX6qnbLBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T05:09:10.616563Z"},"content_sha256":"228a3bf08784f3200b37cd6e5310994b0c3fe52ee72679b5691dab4260ad36b4","schema_version":"1.0","event_id":"sha256:228a3bf08784f3200b37cd6e5310994b0c3fe52ee72679b5691dab4260ad36b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:UDURJ7LHV3I5OFQJPE5YB346JB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Edge growth in graph squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Goff","submitted_at":"2011-12-21T20:58:37Z","abstract_excerpt":"We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow families of graphs, then the square of $G$ has at least $(2-o_d(1))n$ more edges than $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5157","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":"2026-05-18T04:05:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZcAY7yKlLRXxD0uPO4npYNtCJ18tmfkcvMF2tX/cnYOXpE8jHnC1VOl+4n/C5OgonBINkTBzg0XSDa+0TwlfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T05:09:10.616896Z"},"content_sha256":"2139fe47a2b18204b6e3db510830501779d83c4a237118eb799718fc3d31775e","schema_version":"1.0","event_id":"sha256:2139fe47a2b18204b6e3db510830501779d83c4a237118eb799718fc3d31775e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UDURJ7LHV3I5OFQJPE5YB346JB/bundle.json","state_url":"https://pith.science/pith/UDURJ7LHV3I5OFQJPE5YB346JB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UDURJ7LHV3I5OFQJPE5YB346JB/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":"2026-06-25T05:09:10Z","links":{"resolver":"https://pith.science/pith/UDURJ7LHV3I5OFQJPE5YB346JB","bundle":"https://pith.science/pith/UDURJ7LHV3I5OFQJPE5YB346JB/bundle.json","state":"https://pith.science/pith/UDURJ7LHV3I5OFQJPE5YB346JB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UDURJ7LHV3I5OFQJPE5YB346JB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:UDURJ7LHV3I5OFQJPE5YB346JB","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":"a2cebc5352d192e46a8ee4e28a47e3121ef0b811b03e8bb06846ce2763a00210","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T20:58:37Z","title_canon_sha256":"c58da8b5393739f5a1a841945cd13e46707bb31c9b661b2f08dd8256585e5a29"},"schema_version":"1.0","source":{"id":"1112.5157","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.5157","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"arxiv_version","alias_value":"1112.5157v1","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.5157","created_at":"2026-05-18T04:05:48Z"},{"alias_kind":"pith_short_12","alias_value":"UDURJ7LHV3I5","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"UDURJ7LHV3I5OFQJ","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"UDURJ7LH","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:2139fe47a2b18204b6e3db510830501779d83c4a237118eb799718fc3d31775e","target":"graph","created_at":"2026-05-18T04:05:48Z","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 resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow families of graphs, then the square of $G$ has at least $(2-o_d(1))n$ more edges than $G$.","authors_text":"Michael Goff","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T20:58:37Z","title":"Edge growth in graph squares"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5157","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":"sha256:228a3bf08784f3200b37cd6e5310994b0c3fe52ee72679b5691dab4260ad36b4","target":"record","created_at":"2026-05-18T04:05:48Z","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":"a2cebc5352d192e46a8ee4e28a47e3121ef0b811b03e8bb06846ce2763a00210","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-21T20:58:37Z","title_canon_sha256":"c58da8b5393739f5a1a841945cd13e46707bb31c9b661b2f08dd8256585e5a29"},"schema_version":"1.0","source":{"id":"1112.5157","kind":"arxiv","version":1}},"canonical_sha256":"a0e914fd67aed1d71609793b80ef9e48432fd1700e5b3c648caad00552f96acf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a0e914fd67aed1d71609793b80ef9e48432fd1700e5b3c648caad00552f96acf","first_computed_at":"2026-05-18T04:05:48.574214Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:48.574214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8X5sJZ3o6bsmj77Yc/PUWAtGrPaGZm/j1LnA1Y3FWUiyInCew/oBvOo8kXNvbpWldq/xSn+73yLrOlup/NfaAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:48.575003Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.5157","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:228a3bf08784f3200b37cd6e5310994b0c3fe52ee72679b5691dab4260ad36b4","sha256:2139fe47a2b18204b6e3db510830501779d83c4a237118eb799718fc3d31775e"],"state_sha256":"a8afe65898b675c479bcff4854fea471a20c1ad27a51885136df9e942dfe5bcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U2yFgT/R49JZXGSnNx8RRsZoYIJXR55AJbc/NijlO88viTw7bm2CNoBSedD1f2Wbvv/gfNXpPXW7iWK/nX5wBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T05:09:10.618793Z","bundle_sha256":"b5da4409c8946472ceb5017960c5f9fb1f3d9517774ef37f9acfb22e937f45a5"}}