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In this part, we define $V_{n,m}(C_n(R))$ and Type-2 set $T2_{n,m}(C_n(R))$ of $C_n(R)$ and present their properties. We prove that $(V_{n,m}(C_n(R)), \\circ)$ is an Abelian group and $(T2_{n,m}(C_n(R)), \\circ)$ is a subgroup of $(V_{n,m}(C_n(R)), \\circ)$ where $T2_{n,m}(C_n(R))$ = $\\{C_n(R)\\}$ $\\cup$ $\\{C_n(S):$ $C_n(S)$ is Typ-2 isomorphic to $C_n(R)$ w.r.t. $m \\}$ and $(T2_{n,m}(C_n(R)), \\circ)$ is the Type-2 group of $C_n(R)$ w.r.t. $m$. We also present many e","authors_text":"Vilfred Kamalappan","cross_cats":[],"headline":"Circulant graphs related by Type-2 isomorphism form an Abelian group under a defined operation","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T01:24:34Z","title":"A study on Type-2 isomorphic circulant graphs. 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Vilfred,A Study on Isomorphic Properties of Circulant Graphs: Self-complimentary, isomorphism, Cartesian product and factorization, Advances in Science, Technology and Engineering Systems (ASTES) J","work_id":"654d2e75-6b87-4afb-a57d-f5ed0f51fcf7","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Vilfred,A Theory of Cartesian Product and Factorization of Circulant Graphs, Hindawi Pub","work_id":"a8295d8a-29be-4f88-be1d-aff81bbcd01b","year":2013},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Vilfred Kamalappan,A study on Type-2 Isomorphic Circulant Graphs and related Abelian Groups, arXiv: 2012.11372v11 [math.CO] (26 Nov","work_id":"8bcf1163-569c-489a-acfd-b8dd2b05b76b","year":2012}],"snapshot_sha256":"8d68b2c9e716f7e04f7cc7b713b02caf3c2c0b10e5100da8b2bcb16c3f733346"},"source":{"id":"2605.12868","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:45:35.298964Z","id":"03587883-613e-4260-b665-de3b56818bc3","model_set":{"reader":"grok-4.3"},"one_line_summary":"V_{n,m}(C_n(R)) forms an Abelian group under ∘ and T2_{n,m}(C_n(R)) is a subgroup, where T2 collects C_n(R) and all its Type-2 isomorphic copies w.r.t. m.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Circulant graphs related by Type-2 isomorphism form an Abelian group under a defined operation","strongest_claim":"We prove that (V_{n,m}(C_n(R)), ∘) is an Abelian group and (T2_{n,m}(C_n(R)), ∘) is a subgroup of (V_{n,m}(C_n(R)), ∘) where T2_{n,m}(C_n(R)) = {C_n(R)} ∪ {C_n(S): C_n(S) is Type-2 isomorphic to C_n(R) w.r.t. m}.","weakest_assumption":"The operation ∘ is well-defined, associative, and closed on the sets V and T2, and the Type-2 isomorphism relation respects the necessary algebraic identities; this rests on the specific (undefined in abstract) definitions of V, T2, and ∘ introduced in the series."}},"verdict_id":"03587883-613e-4260-b665-de3b56818bc3"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c94c062e3988c23a2b0047f6ed94b9d3eeb89d3075c2a63313c0805a9a3d6a98","target":"record","created_at":"2026-05-18T03:09:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a43b1e97abf3d01572cc08da3f0a619d6a2e850bab95a74d85a6d2f44c22b59d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-13T01:24:34Z","title_canon_sha256":"4b3fa14fdd904e531ede333bbb87bd95ab97f94bc7ba4483bb8cb070061a26da"},"schema_version":"1.0","source":{"id":"2605.12868","kind":"arxiv","version":1}},"canonical_sha256":"49bba91b68166ec881a1454b9fa78211c6ed55911358455ddb8304b97ab13d8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49bba91b68166ec881a1454b9fa78211c6ed55911358455ddb8304b97ab13d8e","first_computed_at":"2026-05-18T03:09:11.407284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:11.407284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mceNCmpRfgU5WI2WTLXhk05O5iYrDonEYApqYE6ttxNhipRb0fD87pJjnl747rxeCAdql07Sbm6WyTb2AgVXAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:11.407995Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12868","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c94c062e3988c23a2b0047f6ed94b9d3eeb89d3075c2a63313c0805a9a3d6a98","sha256:06699c62153322f43fef7b71e332415b5efc991e18225c11b5e12e94e30b67ec","sha256:eb26cce62731e3ab7bf42c8369fe42a16b0250317fdf58c5a42ac717bcb4a219"],"state_sha256":"b8cf67434b4c968dab254626e2d5ca1b78c4f7d944d934a3c83a28e2b83bb78f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4+igIySH0Y7lfqFxdQ4Pp1D4Go3AnloXxii5WsQSVEBv36mrgofrVOQ2c/+KvuAoiSjNb8FkcrGZmLCBRIovBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T19:28:11.307374Z","bundle_sha256":"eb96c2868d899768814a0cb08efe462c74de404be77803b052f1f5e79cf84477"}}