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(i) Family of circulant graphs $C_{432}(R)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3; and (ii) Family of circulant graphs $C_{6750}(S)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3 as well as $m$ = 5. This study is the $8^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \\cite{v2-1}-\\cite{v2-10}."},"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":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.14402","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T05:39:20Z","cross_cats_sorted":[],"title_canon_sha256":"50345caea557d9edcc86483cbf7797e07755b8bf420358a07b139454b546829b","abstract_canon_sha256":"2cfbb2d26806acb38f93ce895bace4064311355c787c466059d697c41209d92f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:07.471152Z","signature_b64":"KLlr/G5GqcZDX14gjhhR9Vyej3KHghrGkV/vgQf0Hu7d5akX5yf3c9x3qxhg3EaHTy4VtX/unebDgM13CQ01CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f94bcdffcd8290b17ad159318bdb3034cb97953b98f11d5aae0ed9c9e24da1b9","last_reissued_at":"2026-05-17T23:39:07.470464Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:07.470464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Study on Type-2 Isomorphic Circulant Graphs: Part 8: $C_{432}(R)$, $C_{6750}(S)$ -- each has 2 types of Type-2 isomorphic circulant graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Families of circulant graphs C_432(R) each admit Type-2 isomorphisms for both m=2 and m=3, and families C_6750(S) do so for both m=3 and m=5.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vilfred Kamalappan","submitted_at":"2026-05-14T05:39:20Z","abstract_excerpt":"In this study, we obtain the following two families of circulant graphs each has Type-2 isomorphic circulant graphs w.r.t. $m$ such that $m$ has more than one value. (i) Family of circulant graphs $C_{432}(R)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 2 as well as $m$ = 3; and (ii) Family of circulant graphs $C_{6750}(S)$, each has isomorphic circulant graphs of Type-2 w.r.t. $m$ = 3 as well as $m$ = 5. This study is the $8^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \\cite{v2-1}-\\cite{v2-10}."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Family of circulant graphs C_432(R), each has isomorphic circulant graphs of Type-2 w.r.t. m = 2 as well as m = 3; and Family of circulant graphs C_6750(S), each has isomorphic circulant graphs of Type-2 w.r.t. m = 3 as well as m = 5.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The specific connection sets R and S are assumed to produce the claimed Type-2 isomorphisms under the definitions established in the author's prior seven papers; this assumption is not independently verified or derived in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Two families of circulant graphs C_432(R) and C_6750(S) each possess Type-2 isomorphic variants for two values of m.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Families of circulant graphs C_432(R) each admit Type-2 isomorphisms for both m=2 and m=3, and families C_6750(S) do so for both m=3 and m=5.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f3071905c62dba11867761f3559aecc52ab211d40ca0cf2078e0c6ddd7f30122"},"source":{"id":"2605.14402","kind":"arxiv","version":1},"verdict":{"id":"596cc171-3985-4f1f-b744-f90b945b1610","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:58:52.160054Z","strongest_claim":"Family of circulant graphs C_432(R), each has isomorphic circulant graphs of Type-2 w.r.t. m = 2 as well as m = 3; and Family of circulant graphs C_6750(S), each has isomorphic circulant graphs of Type-2 w.r.t. m = 3 as well as m = 5.","one_line_summary":"Two families of circulant graphs C_432(R) and C_6750(S) each possess Type-2 isomorphic variants for two values of m.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The specific connection sets R and S are assumed to produce the claimed Type-2 isomorphisms under the definitions established in the author's prior seven papers; this assumption is not independently verified or derived in the abstract.","pith_extraction_headline":"Families of circulant graphs C_432(R) each admit Type-2 isomorphisms for both m=2 and m=3, and families C_6750(S) do so for both m=3 and m=5."},"references":{"count":21,"sample":[{"doi":"","year":1967,"title":"A. 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