{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:FG7KU3YHT5FO4JMRMHDJPUSYRC","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":"3b3e64ca1c9f88d8684e04c063dff9fd58a49898357c51f955d48ded2bd35527","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-08-17T16:14:40Z","title_canon_sha256":"6f839e1f0c2eea4466e9d5cb577d515bb05f7e9a669db7dfe4062bd12d7b31d8"},"schema_version":"1.0","source":{"id":"1808.05915","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1808.05915","created_at":"2026-05-18T00:07:51Z"},{"alias_kind":"arxiv_version","alias_value":"1808.05915v1","created_at":"2026-05-18T00:07:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.05915","created_at":"2026-05-18T00:07:51Z"},{"alias_kind":"pith_short_12","alias_value":"FG7KU3YHT5FO","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"FG7KU3YHT5FO4JMR","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"FG7KU3YH","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:0bdf61a206fc4d6d65c7050c0b8d36522f0a32a43ec633941b25b8351576a811","target":"graph","created_at":"2026-05-18T00:07:51Z","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":"Let a \\neq b be two positive scalars. A Euclidean representation of a simple graph G in R^r is a mapping of the nodes of G into points in R^r such that the squared Euclidean distance between any two points is a if the corresponding nodes are adjacent and b otherwise. A Euclidean representation is spherical if the points lie on an (r-1)-sphere, and is J-spherical if this sphere has radius 1 and a=2 < b. Let dim_E(G), dim_S(G) and dim_J(G) denote, respectively, the smallest dimension r for which G admits a Euclidean, spherical and J-spherical representation.\n  In this paper, we extend and simpli","authors_text":"A. Y. Alfakih","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-08-17T16:14:40Z","title":"On Representations of Graphs as Two-Distance Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05915","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:b962410ae42df7991f502ee1329ec4f330ea15045a50b5ec717cabe528c35690","target":"record","created_at":"2026-05-18T00:07:51Z","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":"3b3e64ca1c9f88d8684e04c063dff9fd58a49898357c51f955d48ded2bd35527","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-08-17T16:14:40Z","title_canon_sha256":"6f839e1f0c2eea4466e9d5cb577d515bb05f7e9a669db7dfe4062bd12d7b31d8"},"schema_version":"1.0","source":{"id":"1808.05915","kind":"arxiv","version":1}},"canonical_sha256":"29beaa6f079f4aee259161c697d2588884bbe4e4b92312e42d9b2d7a67995149","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29beaa6f079f4aee259161c697d2588884bbe4e4b92312e42d9b2d7a67995149","first_computed_at":"2026-05-18T00:07:51.659620Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:07:51.659620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mNIBJuDNdpAgJPWZCtFWH3CPYpIB+NHDoHX6/qrINKLqK0EY2IsdwfCK79aBGTm0wvAXQfIUJEFx/gfVXzUICw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:07:51.660334Z","signed_message":"canonical_sha256_bytes"},"source_id":"1808.05915","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b962410ae42df7991f502ee1329ec4f330ea15045a50b5ec717cabe528c35690","sha256:0bdf61a206fc4d6d65c7050c0b8d36522f0a32a43ec633941b25b8351576a811"],"state_sha256":"fecd522225c6fefb6c1314433666462ae191febf785d420e27ddb90f95baca84"}