{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AYROEHHX4LTMTAUHK2XMG6NP7O","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":"6fa1ce733c6b8f1d708c98b003f3748b8b80aabc67dd180a67bc4ea6cc79622c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-30T17:38:17Z","title_canon_sha256":"6f5f3b347b0dd5986b377bef33b8a56bdc04ae19d66f9348910ce2c1487228ea"},"schema_version":"1.0","source":{"id":"1111.7246","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.7246","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"arxiv_version","alias_value":"1111.7246v1","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.7246","created_at":"2026-05-18T04:07:18Z"},{"alias_kind":"pith_short_12","alias_value":"AYROEHHX4LTM","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AYROEHHX4LTMTAUH","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AYROEHHX","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:276fe01b143acb539873d49676945519366eb36567679920103145b66846c8c1","target":"graph","created_at":"2026-05-18T04:07:18Z","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 provide a complete description of important geometric invariants of the Laplacian lattice of a multigraph under the distance function induced by a regular simplex, namely Voronoi Diagram, Delaunay Triangulation, Delaunay Polytope and its combinatorial structure, Shortest Vectors, Covering and Packing Radius. We use this information to obtain the following results: i. Every multigraph defines a Delaunay triangulation of its Laplacian lattice and this Delaunay triangulation contains complete information of the multigraph up to isomorphism. ii. The number of multigraphs with a given Laplacian ","authors_text":"Madhusudan Manjunath","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-30T17:38:17Z","title":"The Laplacian lattice of a graph under a simplicial distance function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.7246","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:b9e0e27cd2735019164a611470f08bbbb4d09388ae5c2bc772b421bb9e25c13b","target":"record","created_at":"2026-05-18T04:07:18Z","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":"6fa1ce733c6b8f1d708c98b003f3748b8b80aabc67dd180a67bc4ea6cc79622c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-11-30T17:38:17Z","title_canon_sha256":"6f5f3b347b0dd5986b377bef33b8a56bdc04ae19d66f9348910ce2c1487228ea"},"schema_version":"1.0","source":{"id":"1111.7246","kind":"arxiv","version":1}},"canonical_sha256":"0622e21cf7e2e6c9828756aec379affb9b451c5872bc5dcaa8283c5df203bfb5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0622e21cf7e2e6c9828756aec379affb9b451c5872bc5dcaa8283c5df203bfb5","first_computed_at":"2026-05-18T04:07:18.331408Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:18.331408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nx7L1IP4RKmlNlJzkmlqCm/4DMpEI7DmxWiQorjYMwHr+vntkqdTU7sYcv9Clt50QxOKHh0WjO9agwW/EfyEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:18.332100Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.7246","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9e0e27cd2735019164a611470f08bbbb4d09388ae5c2bc772b421bb9e25c13b","sha256:276fe01b143acb539873d49676945519366eb36567679920103145b66846c8c1"],"state_sha256":"969bc0e214fdf7e1b8d9fa3f5da3d4d0e3f4921475a5ec029e5d665819862ccb"}