{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:UJGP36YOIH64DIXZTNJIF6MYOA","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":"ab3f54eea3036ba473142262f364fd154e8608137898bc37adb8b641d7e40693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-07T14:18:29Z","title_canon_sha256":"bfd06d303be3542e08c18490c11d8d8b4c46948d32503b665217add6debeaa12"},"schema_version":"1.0","source":{"id":"1903.02929","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.02929","created_at":"2026-05-17T23:51:41Z"},{"alias_kind":"arxiv_version","alias_value":"1903.02929v2","created_at":"2026-05-17T23:51:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.02929","created_at":"2026-05-17T23:51:41Z"},{"alias_kind":"pith_short_12","alias_value":"UJGP36YOIH64","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_16","alias_value":"UJGP36YOIH64DIXZ","created_at":"2026-05-18T12:33:30Z"},{"alias_kind":"pith_short_8","alias_value":"UJGP36YO","created_at":"2026-05-18T12:33:30Z"}],"graph_snapshots":[{"event_id":"sha256:ee474e35dc1a5a9b80b1d5b915b0e0f75733b41b86073b02aaeed2e1aea94db8","target":"graph","created_at":"2026-05-17T23:51:41Z","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 rank $r(G)$ of a graph $G$ is the rank of its adjacency matrix $A(G)$ and the nullity $\\eta(G)$ of $G$ is the multiplicity of $0$ as an eigenvalue of $A(G)$. In this paper, we prove that if $G$ is a connected graph of order $n$ with rank $r$, then $G$ contains a nonsingular connected induced subgraph of order $r$. As an application of the result, we completely solve the following problem posed by Zhou, Wong and Sun in [Linear Algebra and its Applications, 555 (2018) 314-320]: Let $G$ be a connected graph of order $n$ with nullity $\\eta(G)$ and the maximum degree $\\Delta$. Then $$\\eta(G)\\le","authors_text":"Jiming Guo, Zhiwen Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-07T14:18:29Z","title":"On the rank (nullity) of a connected graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.02929","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":"sha256:5cceaa6232155a2dfc95193d3a5d77e502de15efa8af3e7550d201673b7cfa61","target":"record","created_at":"2026-05-17T23:51:41Z","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":"ab3f54eea3036ba473142262f364fd154e8608137898bc37adb8b641d7e40693","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-07T14:18:29Z","title_canon_sha256":"bfd06d303be3542e08c18490c11d8d8b4c46948d32503b665217add6debeaa12"},"schema_version":"1.0","source":{"id":"1903.02929","kind":"arxiv","version":2}},"canonical_sha256":"a24cfdfb0e41fdc1a2f99b5282f998702f4921ca02709766bfe499aeb55c0227","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a24cfdfb0e41fdc1a2f99b5282f998702f4921ca02709766bfe499aeb55c0227","first_computed_at":"2026-05-17T23:51:41.515628Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:41.515628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eMe5Nj8IxvifH0cGQZnJDLH9YVZYmco+gkSh64/XlbcA3Cu5tY14q7o1MWtZzxQ9nBkYumHpRI8c3JVEfJQYCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:41.516341Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.02929","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5cceaa6232155a2dfc95193d3a5d77e502de15efa8af3e7550d201673b7cfa61","sha256:ee474e35dc1a5a9b80b1d5b915b0e0f75733b41b86073b02aaeed2e1aea94db8"],"state_sha256":"78db5c6d2cfddfdb511d9659534daef34832ca13635e7c3ee877b05c9fa4e177"}