{"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. Adam,Research problem 2-10, J. Combinatorial Theory,3(1967), 393","work_id":"7a43d6b0-2dfb-4412-99ad-96186a8a7489","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1982,"title":"J. A. Bondy and U. S. R. Murty,Graph Theory with Applications,5 th Edi., Elsevier Sci. Publ. Co., New York, 1982","work_id":"5fcc58de-52d9-4670-825b-e6b132effaad","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1979,"title":"P. J. Davis,Circulant Matrices,Wiley, New York, 1979","work_id":"0f76597d-55e1-4004-87ef-8efe1734b290","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"West,Introduction to Graph Theory,2 ed Edi., Pearson Education (Singapore) Pvt","work_id":"cad7d710-39d0-4154-96c0-117798e78ec1","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1970,"title":"B. Elspas and J. Turner,Graphs with circulant adjacency matrices, J. Combinatorial Theory,9(1970), 297-307","work_id":"d1c668b3-b8ac-436d-9047-2408b9c6f3ec","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":21,"snapshot_sha256":"3c828d45fe615bc80f52b71e5d470906c8f5bb07a007fc6198bdc9a232f59de2","internal_anchors":1},"formal_canon":{"evidence_count":3,"snapshot_sha256":"17ef3e6fac673831b1072beab57ac1e954e3e5f6dd414369a991db8bddc9a2a9"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}