{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:G7Y4V4JCBUWFRQWUXNOJAJBGGA","short_pith_number":"pith:G7Y4V4JC","canonical_record":{"source":{"id":"1206.3409","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-15T09:24:48Z","cross_cats_sorted":[],"title_canon_sha256":"b94978457599881e0f863da2dd8d8fd82f3fadfd7c38d451da4f6d376fce43fb","abstract_canon_sha256":"3c2393d386555a5017f33f415517e6e2136517190ce4a02925e5677403a55a5c"},"schema_version":"1.0"},"canonical_sha256":"37f1caf1220d2c58c2d4bb5c902426301d18da9478d82a1d64cae9707d15070f","source":{"kind":"arxiv","id":"1206.3409","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.3409","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"arxiv_version","alias_value":"1206.3409v1","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.3409","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"pith_short_12","alias_value":"G7Y4V4JCBUWF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G7Y4V4JCBUWFRQWU","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G7Y4V4JC","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:G7Y4V4JCBUWFRQWUXNOJAJBGGA","target":"record","payload":{"canonical_record":{"source":{"id":"1206.3409","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-15T09:24:48Z","cross_cats_sorted":[],"title_canon_sha256":"b94978457599881e0f863da2dd8d8fd82f3fadfd7c38d451da4f6d376fce43fb","abstract_canon_sha256":"3c2393d386555a5017f33f415517e6e2136517190ce4a02925e5677403a55a5c"},"schema_version":"1.0"},"canonical_sha256":"37f1caf1220d2c58c2d4bb5c902426301d18da9478d82a1d64cae9707d15070f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:51:03.644179Z","signature_b64":"bnFBYyVvbKm1wxp1yyXH5tBN65VBRV9sjb1JEwZGxgx75naReajQVrSz86JqzTm8Aqh+SJySzGkuvPDv3sXaAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37f1caf1220d2c58c2d4bb5c902426301d18da9478d82a1d64cae9707d15070f","last_reissued_at":"2026-05-18T00:51:03.643684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:51:03.643684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.3409","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":"2026-05-18T00:51:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZNqet3zz9aY7hYRJUM02xywRulngQ176J48u6jseXehvPUNzKn68gpuGThdO8YNLTPjsjniAYIbXIoN56MSTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:17:08.209070Z"},"content_sha256":"d193089832d2899bcb14eb4ba4312209bded883ff8ae12a8ae6c8996560f9e46","schema_version":"1.0","event_id":"sha256:d193089832d2899bcb14eb4ba4312209bded883ff8ae12a8ae6c8996560f9e46"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:G7Y4V4JCBUWFRQWUXNOJAJBGGA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on the minimum skew rank of a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Zhou, Yanna Wang","submitted_at":"2012-06-15T09:24:48Z","abstract_excerpt":"The minimum skew rank $mr^{-}(\\mathbb{F},G)$ of a graph $G$ over a field $\\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\\mathbb{F}$, whose ($i$,$j$)-entry (for $i\\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\\mathbb{F}$ such that $mr^{-}(\\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\\mathbb{F}$, and an extending of an existing result to sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3409","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":"2026-05-18T00:51:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"skyskXpvkZAwS2d/qDuEZer3ZYKXuCPJYmbFP8EMGymYrjn26cI2OWahJlotMTAi3TK3el86VZvaM5iLtc4YCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T15:17:08.209428Z"},"content_sha256":"3a772710a5cc524fec7f83024e9f7d88c1deb863419de9086b81319ccac407bf","schema_version":"1.0","event_id":"sha256:3a772710a5cc524fec7f83024e9f7d88c1deb863419de9086b81319ccac407bf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/bundle.json","state_url":"https://pith.science/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/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":"2026-06-10T15:17:08Z","links":{"resolver":"https://pith.science/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA","bundle":"https://pith.science/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/bundle.json","state":"https://pith.science/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G7Y4V4JCBUWFRQWUXNOJAJBGGA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:G7Y4V4JCBUWFRQWUXNOJAJBGGA","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":"3c2393d386555a5017f33f415517e6e2136517190ce4a02925e5677403a55a5c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-15T09:24:48Z","title_canon_sha256":"b94978457599881e0f863da2dd8d8fd82f3fadfd7c38d451da4f6d376fce43fb"},"schema_version":"1.0","source":{"id":"1206.3409","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.3409","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"arxiv_version","alias_value":"1206.3409v1","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.3409","created_at":"2026-05-18T00:51:03Z"},{"alias_kind":"pith_short_12","alias_value":"G7Y4V4JCBUWF","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G7Y4V4JCBUWFRQWU","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G7Y4V4JC","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:3a772710a5cc524fec7f83024e9f7d88c1deb863419de9086b81319ccac407bf","target":"graph","created_at":"2026-05-18T00:51:03Z","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":"The minimum skew rank $mr^{-}(\\mathbb{F},G)$ of a graph $G$ over a field $\\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\\mathbb{F}$, whose ($i$,$j$)-entry (for $i\\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\\mathbb{F}$ such that $mr^{-}(\\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\\mathbb{F}$, and an extending of an existing result to sho","authors_text":"Bo Zhou, Yanna Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-15T09:24:48Z","title":"A note on the minimum skew rank of a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.3409","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:d193089832d2899bcb14eb4ba4312209bded883ff8ae12a8ae6c8996560f9e46","target":"record","created_at":"2026-05-18T00:51:03Z","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":"3c2393d386555a5017f33f415517e6e2136517190ce4a02925e5677403a55a5c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-06-15T09:24:48Z","title_canon_sha256":"b94978457599881e0f863da2dd8d8fd82f3fadfd7c38d451da4f6d376fce43fb"},"schema_version":"1.0","source":{"id":"1206.3409","kind":"arxiv","version":1}},"canonical_sha256":"37f1caf1220d2c58c2d4bb5c902426301d18da9478d82a1d64cae9707d15070f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37f1caf1220d2c58c2d4bb5c902426301d18da9478d82a1d64cae9707d15070f","first_computed_at":"2026-05-18T00:51:03.643684Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:03.643684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bnFBYyVvbKm1wxp1yyXH5tBN65VBRV9sjb1JEwZGxgx75naReajQVrSz86JqzTm8Aqh+SJySzGkuvPDv3sXaAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:03.644179Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.3409","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d193089832d2899bcb14eb4ba4312209bded883ff8ae12a8ae6c8996560f9e46","sha256:3a772710a5cc524fec7f83024e9f7d88c1deb863419de9086b81319ccac407bf"],"state_sha256":"7e0967eede691717724b11706ec4d98da559ead064c05463c5caf16635393c20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fe6cIlRcKYohJGwimiaR8zrS9yGLGDqcjwDUwJPYC9zS7hxw2FRfv1+ghOrZLFr47BpMh3nb/fZaGxyXFDQUCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T15:17:08.211422Z","bundle_sha256":"bff724e73889ba807b13c621ef0698c5439fc982950882fe32d5526584728a7c"}}