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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.05915","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2018-08-17T16:14:40Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6f839e1f0c2eea4466e9d5cb577d515bb05f7e9a669db7dfe4062bd12d7b31d8","abstract_canon_sha256":"3b3e64ca1c9f88d8684e04c063dff9fd58a49898357c51f955d48ded2bd35527"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:51.660334Z","signature_b64":"mNIBJuDNdpAgJPWZCtFWH3CPYpIB+NHDoHX6/qrINKLqK0EY2IsdwfCK79aBGTm0wvAXQfIUJEFx/gfVXzUICw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"29beaa6f079f4aee259161c697d2588884bbe4e4b92312e42d9b2d7a67995149","last_reissued_at":"2026-05-18T00:07:51.659620Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:51.659620Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Representations of Graphs as Two-Distance Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.MG","authors_text":"A. 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