{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JBE43R36LUXMHGVE7Z634USWB4","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":"2ffd8cdf8d060ae877c40d02a5d68981f509b55e74b44ba9116e77ea3b3493b8","cross_cats_sorted":["cs.CG","cs.DM","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-01-05T13:10:27Z","title_canon_sha256":"89714dbb496881d22968fc3a0e1b18ed6b9146b9a2c2cabfd2379aa6bea1d69b"},"schema_version":"1.0","source":{"id":"1201.1162","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.1162","created_at":"2026-05-18T04:05:05Z"},{"alias_kind":"arxiv_version","alias_value":"1201.1162v1","created_at":"2026-05-18T04:05:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.1162","created_at":"2026-05-18T04:05:05Z"},{"alias_kind":"pith_short_12","alias_value":"JBE43R36LUXM","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JBE43R36LUXMHGVE","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JBE43R36","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:4c6037a16d8e42f0e0401223ffebbcfe3f3a273d6e7fffc86f044b0c46fd76eb","target":"graph","created_at":"2026-05-18T04:05:05Z","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 introduce the index i(v) = 1 - X(S(v)) for critical points of a locally injective function f on the vertex set V of a simple graph G=(V,E). Here S(v) = {w in E | (v,w) in E, f(w)-f(v)<0} is the subgraph of the unit sphere at v in G. It is the exit set of the gradient vector field. We prove that the sum of i(v) over V is always is equal to the Euler characteristic X(G) of the graph G. This is a discrete Poincare-Hopf theorem in a discrete Morse setting. It allows to compute X(G) for large graphs for which other methods become impractical.","authors_text":"Oliver Knill","cross_cats":["cs.CG","cs.DM","math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-01-05T13:10:27Z","title":"A graph theoretical Poincare-Hopf Theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.1162","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:e28386c16cd0fa9dd78336172cdc4cb44c8f30c9a9c49356d72b1a2d4006b180","target":"record","created_at":"2026-05-18T04:05:05Z","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":"2ffd8cdf8d060ae877c40d02a5d68981f509b55e74b44ba9116e77ea3b3493b8","cross_cats_sorted":["cs.CG","cs.DM","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-01-05T13:10:27Z","title_canon_sha256":"89714dbb496881d22968fc3a0e1b18ed6b9146b9a2c2cabfd2379aa6bea1d69b"},"schema_version":"1.0","source":{"id":"1201.1162","kind":"arxiv","version":1}},"canonical_sha256":"4849cdc77e5d2ec39aa4fe7dbe52560f123d66acefb5ffb193491b2190e8cab6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4849cdc77e5d2ec39aa4fe7dbe52560f123d66acefb5ffb193491b2190e8cab6","first_computed_at":"2026-05-18T04:05:05.907916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:05:05.907916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wPhRVDeEjUUzuP3gELtClhfpKjjs0HYc7Fs54fZxtzfAxFb0RSt+hMhkWs07JBYGSJ6NwT9mGzGDlXglCHEdBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:05:05.908629Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.1162","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e28386c16cd0fa9dd78336172cdc4cb44c8f30c9a9c49356d72b1a2d4006b180","sha256:4c6037a16d8e42f0e0401223ffebbcfe3f3a273d6e7fffc86f044b0c46fd76eb"],"state_sha256":"eba1f4c4705b9e63dcb424dc223fd5ec8a524a4f99347afeea9ccf707e5a863e"}