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They are Cayley graphs on the additive group of $\\mathbb{Z}[\\zeta_m]/A$, with connection sets $\\{\\pm (\\zeta_m^i + A): 0 \\le i \\le m-1\\}$ and $\\{\\pm (\\zeta_m^i + A): 0 \\le i \\le \\phi(m) - 1\\}$, respectively, where $\\zeta_m$ ($m \\ge 2$) is an $m$th primitive root of unity, $A$ a nonzero ideal of $\\mathbb{Z}[\\zeta_m]$, and $\\phi$ Euler's totient function. We call them the $m$th cyclotomic graph and the second kind $m$th cyclotomic graph, and denote them by $G_{m}(A)$ and $G^*_{m}(A)$, respectively. We give a necessary and suffi"},"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":"1502.03272","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-02-11T11:56:46Z","cross_cats_sorted":[],"title_canon_sha256":"0071964a6eaf9987b9635a763a1dbf52829e074e365610a7e19dce5879e7f94e","abstract_canon_sha256":"a98c51687c963c4894d27000ab89e65f6d0bf5d7b3ddd8b4137aefbdc1b823f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:47.799049Z","signature_b64":"iMo6m8LfOr4CdTK/DZI3GpeTBqsnMyBaLFLPyeuWmu9uKH+xQEQttgy5eYRAU1NOp84kAmWuiHUYILKFlMjCAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"296d8e67549b274460ca05d40e82a6c388d0d60304b3492017876533a4f60f87","last_reissued_at":"2026-05-18T00:04:47.798376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:47.798376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cyclotomic graphs and perfect codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sanming Zhou","submitted_at":"2015-02-11T11:56:46Z","abstract_excerpt":"We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\\mathbb{Z}[\\zeta_m]/A$, with connection sets $\\{\\pm (\\zeta_m^i + A): 0 \\le i \\le m-1\\}$ and $\\{\\pm (\\zeta_m^i + A): 0 \\le i \\le \\phi(m) - 1\\}$, respectively, where $\\zeta_m$ ($m \\ge 2$) is an $m$th primitive root of unity, $A$ a nonzero ideal of $\\mathbb{Z}[\\zeta_m]$, and $\\phi$ Euler's totient function. We call them the $m$th cyclotomic graph and the second kind $m$th cyclotomic graph, and denote them by $G_{m}(A)$ and $G^*_{m}(A)$, respectively. 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