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The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1 \\leq u \\leq n-1$ and $\\gcd(u,n) = 1$. In this paper, we classify all $X_n$ admitting pretty good fractional. We also classify all $X_n$ that admit fractional revival. It turns out that"},"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":"2508.18068","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-08-25T14:26:06Z","cross_cats_sorted":[],"title_canon_sha256":"be8e1b0f56e41ba2596aaca4e78a96c21bb0f8ddca4b3d6ccfcb06fda45d07ad","abstract_canon_sha256":"5138099396dd95f1fe6fc3781237562b4e0fb11b1c7c22cd56dd121495d36992"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:58:53.138199Z","signature_b64":"vqQbNyXuWJUKS7T6gIqAbTGMRMMLlFoUvKSndbnE4/foAHQnpG4rZI4WH4WA2eKRWCqwoG/OD09Bmv8RFIxEAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8380f4ef0b10efac94c142aee474b34b0d025ec657daf78e651bc8de199d9e9","last_reissued_at":"2026-07-05T11:58:53.137726Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:58:53.137726Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"State Transfer on Unitary Cayley Graphs and Quadratic Unitary Cayley Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Akash Kalita, Bikash Bhattacharjya","submitted_at":"2025-08-25T14:26:06Z","abstract_excerpt":"The unitary Cayley graph, denoted $X_n$, is the graph with vertex set ${\\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u$ for some $u$ with $1 \\leq u \\leq n-1$ and $\\gcd(u,n) = 1$. The quadratic unitary Cayley graph, denoted $G_n$, is the graph with vertex set ${\\mathbb{Z}}_n$ such that two distinct vertices $a$ and $b$ are adjacent if $a-b=u^2$ or $a-b=-u^2$ for some $u$ with $1 \\leq u \\leq n-1$ and $\\gcd(u,n) = 1$. In this paper, we classify all $X_n$ admitting pretty good fractional. We also classify all $X_n$ that admit fractional revival. 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