{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:LDRMFWZHNIFOCWQZSX774MMKVJ","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":"64e55659ee712df109c50af52af70f8385c60ea51549cf258daf69c08d8313b0","cross_cats_sorted":["cs.DM","math.MG","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-01T05:39:33Z","title_canon_sha256":"20585e4d49373348b4c6cc1466da544bed6abea33597f7a75e12d2953933ae53"},"schema_version":"1.0","source":{"id":"1112.0088","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0088","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0088v4","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0088","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"LDRMFWZHNIFO","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"LDRMFWZHNIFOCWQZ","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"LDRMFWZH","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:dac065804b7ab165b1c08afb985f8b375632a49d683a624449934d09c5c82316","target":"graph","created_at":"2026-05-18T03:46:03Z","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 walk distances in graphs are defined as the result of appropriate transformations of the $\\sum_{k=0}^\\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices o","authors_text":"Pavel Chebotarev, R. Balaji, R. B. Bapat","cross_cats":["cs.DM","math.MG","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-01T05:39:33Z","title":"Simple expressions for the long walk distance"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0088","kind":"arxiv","version":4},"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":"sha256:5d14d04b606cc9a572fba1c3ad1e19c5200102bd78cf442f69d23048b1de9f30","target":"record","created_at":"2026-05-18T03:46:03Z","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":"64e55659ee712df109c50af52af70f8385c60ea51549cf258daf69c08d8313b0","cross_cats_sorted":["cs.DM","math.MG","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-12-01T05:39:33Z","title_canon_sha256":"20585e4d49373348b4c6cc1466da544bed6abea33597f7a75e12d2953933ae53"},"schema_version":"1.0","source":{"id":"1112.0088","kind":"arxiv","version":4}},"canonical_sha256":"58e2c2db276a0ae15a1995fffe318aaa745a4bdb76e848896add8eb2bf0a749f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"58e2c2db276a0ae15a1995fffe318aaa745a4bdb76e848896add8eb2bf0a749f","first_computed_at":"2026-05-18T03:46:03.663909Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:03.663909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8E5Gr9TBgzKlrG+69lrzDcVWwVTU5wdXOdIuETP5mjKLK2FvKI5BLzM83eCQoSh3vQhbJtKbPUN0y286Hd+KBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:03.664481Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.0088","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d14d04b606cc9a572fba1c3ad1e19c5200102bd78cf442f69d23048b1de9f30","sha256:dac065804b7ab165b1c08afb985f8b375632a49d683a624449934d09c5c82316"],"state_sha256":"425cce62441c908df0a8f7b9c5af245bde1fd3da50b335077dd4f1eb30728777"}