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Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\\mathbb{CAY}(R)$ as well. We also study $Reg(\\"},"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":"1305.0601","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-05-03T00:25:24Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"bd55a09b6ced2e1718a8262992ad921b9ba43f21aac57ac3fafdd1e620dce6e0","abstract_canon_sha256":"38e0762a483e87f7c48bc3b67a014d97a15c2104c0cb9f80be1265d661d9e9be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:33.239865Z","signature_b64":"mVUq8vzrvSrUeRs903tiB9Qxji2qC63PyHaHkvOPN8klx7tdX5akGhu5qnPmMc2jqn+/X23JlPj/ow8iukZRCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7dfbb4d00f1f0c83d5a4f572067a39c7ce5235089f71c7b956d440dc120cacdc","last_reissued_at":"2026-05-18T03:26:33.239437Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:33.239437Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Cayley graph of a commutative ring with respect to its zero-divisors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Ghodratollah Aalipour, Saieed Akbari","submitted_at":"2013-05-03T00:25:24Z","abstract_excerpt":"Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. 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