{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:IFKLAHWXHC5NRFYQ65AR5Q6Z6G","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":"51f7e3105f41d10e1c64298e217ef8d68a065f3cea77e434851a72c19b1c3cb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-01T14:21:51Z","title_canon_sha256":"8190578f0511ec1ba2141d15c0e2de9510a58c7d12708cdb7903045d52f6498c"},"schema_version":"1.0","source":{"id":"1603.00282","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.00282","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"arxiv_version","alias_value":"1603.00282v1","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.00282","created_at":"2026-05-18T01:19:45Z"},{"alias_kind":"pith_short_12","alias_value":"IFKLAHWXHC5N","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_16","alias_value":"IFKLAHWXHC5NRFYQ","created_at":"2026-05-18T12:30:22Z"},{"alias_kind":"pith_short_8","alias_value":"IFKLAHWX","created_at":"2026-05-18T12:30:22Z"}],"graph_snapshots":[{"event_id":"sha256:1189505dfb614c6f4bba7b1fb64dbc413b53712965cb6f8f4d8db7b65dbb9319","target":"graph","created_at":"2026-05-18T01:19:45Z","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 vertex PI index $PI(G) = \\sum_{xy \\in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the largest and smallest vertex PI indices among all the cacti. As a consequence, we obtain the sharp bound","authors_text":"Bing Wei, Chunxiang Wang, Shaohui Wang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-01T14:21:51Z","title":"Cacti with Extremal PI Index"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00282","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:c9bd8f86931a97e3f2eced3f9af46f83e4a32bcc64ed7029432ee10d1e6ad7db","target":"record","created_at":"2026-05-18T01:19:45Z","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":"51f7e3105f41d10e1c64298e217ef8d68a065f3cea77e434851a72c19b1c3cb1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-03-01T14:21:51Z","title_canon_sha256":"8190578f0511ec1ba2141d15c0e2de9510a58c7d12708cdb7903045d52f6498c"},"schema_version":"1.0","source":{"id":"1603.00282","kind":"arxiv","version":1}},"canonical_sha256":"4154b01ed738bad89710f7411ec3d9f19696e503ead30d81025b5e6d46c9abc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4154b01ed738bad89710f7411ec3d9f19696e503ead30d81025b5e6d46c9abc5","first_computed_at":"2026-05-18T01:19:45.443926Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:45.443926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CCMWfJ0v5xj+gJ5jPdeq5CMNTfM+5YYHgCrrCAuoo7stAUYHtvI+B7MxUykQ4HLOBwaVP+jrRw83UDCKJ+Z5BA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:45.444427Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.00282","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c9bd8f86931a97e3f2eced3f9af46f83e4a32bcc64ed7029432ee10d1e6ad7db","sha256:1189505dfb614c6f4bba7b1fb64dbc413b53712965cb6f8f4d8db7b65dbb9319"],"state_sha256":"ea0fa51d5b07ab627c96809dbd1be91e4793d0201e34375f43c8bf31c8cb8267"}