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We define the quadratic unitary Cayley graph of $R$, denoted by $\\mathcal{G}_R$, to be the Cayley graph on the additive group of $R$ with respect to $T_R$; that is, $\\mathcal{G}_R$ has vertex set $R$ such that $x, y \\in R$ are adjacent if and only if $x-y\\in T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1\\times R_2\\times\\cdots\\t"},"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":"1504.02934","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-12T05:48:51Z","cross_cats_sorted":[],"title_canon_sha256":"551c8322e0b9e39983c89db68f232106ce7f74945a4d4c10c319669430f32ade","abstract_canon_sha256":"52fe30a929e06bba377470d906013ac66a80045746d2e9645f967b0a40a890c1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:59.620542Z","signature_b64":"SLh9/MNXFw64z6/bxa4LHMS9kRu4xFX2ixn+VaEiN6HdhFr6J0OwBnIhZVIh5zFpwvplbk62vUsCWO2RXWenCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb0fa96b997a546a320d8d4f3802ae0b6e1776bcb69787c2e2e960ed55f2cd4b","last_reissued_at":"2026-05-18T02:18:59.620032Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:59.620032Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic unitary Cayley graphs of finite commutative rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sanming Zhou, Xiaogang Liu","submitted_at":"2015-04-12T05:48:51Z","abstract_excerpt":"The purpose of this paper is to study spectral properties of a family of Cayley graphs on finite commutative rings. Let $R$ be such a ring and $R^\\times$ its set of units. Let $Q_R=\\{u^2: u\\in R^\\times\\}$ and $T_R=Q_R\\cup(-Q_R)$. We define the quadratic unitary Cayley graph of $R$, denoted by $\\mathcal{G}_R$, to be the Cayley graph on the additive group of $R$ with respect to $T_R$; that is, $\\mathcal{G}_R$ has vertex set $R$ such that $x, y \\in R$ are adjacent if and only if $x-y\\in T_R$. It is well known that any finite commutative ring $R$ can be decomposed as $R=R_1\\times R_2\\times\\cdots\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02934","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.02934","created_at":"2026-05-18T02:18:59.620104+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.02934v1","created_at":"2026-05-18T02:18:59.620104+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.02934","created_at":"2026-05-18T02:18:59.620104+00:00"},{"alias_kind":"pith_short_12","alias_value":"7MH2S24ZPJKG","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7MH2S24ZPJKGUMQN","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7MH2S24Z","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN","json":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN.json","graph_json":"https://pith.science/api/pith-number/7MH2S24ZPJKGUMQNRVHTQAVOBN/graph.json","events_json":"https://pith.science/api/pith-number/7MH2S24ZPJKGUMQNRVHTQAVOBN/events.json","paper":"https://pith.science/paper/7MH2S24Z"},"agent_actions":{"view_html":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN","download_json":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN.json","view_paper":"https://pith.science/paper/7MH2S24Z","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.02934&json=true","fetch_graph":"https://pith.science/api/pith-number/7MH2S24ZPJKGUMQNRVHTQAVOBN/graph.json","fetch_events":"https://pith.science/api/pith-number/7MH2S24ZPJKGUMQNRVHTQAVOBN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN/action/storage_attestation","attest_author":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN/action/author_attestation","sign_citation":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN/action/citation_signature","submit_replication":"https://pith.science/pith/7MH2S24ZPJKGUMQNRVHTQAVOBN/action/replication_record"}},"created_at":"2026-05-18T02:18:59.620104+00:00","updated_at":"2026-05-18T02:18:59.620104+00:00"}