{"paper":{"title":"A study on Type-2 isomorphic circulant graphs. PART 9: Computer programs to show Type-1 $\\&$ -2 isomorphic circulant graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Computer programs generate families of Type-2 isomorphic circulant graphs for m values 2, 3, 5, 7 and odd primes p.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Vilfred Kamalappan, Wilson Peraprakash","submitted_at":"2026-05-13T21:46:47Z","abstract_excerpt":"Elspas and Turner \\cite{eltu} raised a question on the isomorphism of $C_{16}(1,3,7)$ and $C_{16}(2,3,5)$ and Vilfred \\cite{v96} gave its answer by defining Type-2 isomorphism of $C_n(R)$ w.r.t. $m$ $\\ni$ $m$ = $\\gcd(n, r) > 1$, $r\\in R$ and $r,n\\in\\mathbb{N}$ and studied such graphs for $m$ = 2 in \\cite{v13,v20}. But obtaining Type-2 isomorphic circulant graphs is not easy. Using a $C^{++}$ computer program, the authors obtained families of Type-2 isomorphic $C_{n}(R)$ w.r.t. $m$ = 2,3,5,7 for $n\\in\\mathbb{N}$ as well as $C_{np^3}(R)$ w.r.t. $m$ = $p$ for $n\\in\\mathbb{N}$ and $p$ is an odd pr"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Using a C++ computer program, the authors obtained families of Type-2 isomorphic C_n(R) w.r.t. m = 2,3,5,7 for n in natural numbers as well as C_{np^3}(R) w.r.t. m = p for odd prime p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The programs correctly implement the definitions of Type-1 and Type-2 isomorphism from the cited prior papers without coding errors or missed cases.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Computer programs are supplied to generate families of Type-2 isomorphic circulant graphs C_n(R) for m=2,3,5,7 and to demonstrate how Type-1 and Type-2 isomorphisms occur.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Computer programs generate families of Type-2 isomorphic circulant graphs for m values 2, 3, 5, 7 and odd primes p.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"3c4b7273fb4780b744e493870b39e1c48eb7222fbcb3f194a5c11178bcaa4256"},"source":{"id":"2605.14140","kind":"arxiv","version":1},"verdict":{"id":"f0e42807-d5e7-4d54-82d7-7112bc309a50","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:11:18.171518Z","strongest_claim":"Using a C++ computer program, the authors obtained families of Type-2 isomorphic C_n(R) w.r.t. m = 2,3,5,7 for n in natural numbers as well as C_{np^3}(R) w.r.t. m = p for odd prime p.","one_line_summary":"Computer programs are supplied to generate families of Type-2 isomorphic circulant graphs C_n(R) for m=2,3,5,7 and to demonstrate how Type-1 and Type-2 isomorphisms occur.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The programs correctly implement the definitions of Type-1 and Type-2 isomorphism from the cited prior papers without coding errors or missed cases.","pith_extraction_headline":"Computer programs generate families of Type-2 isomorphic circulant graphs for m values 2, 3, 5, 7 and odd primes p."},"references":{"count":25,"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":2013,"title":"Bjarne Stroustrup,The C++ Programming Language, Addison-Wesley, 2013","work_id":"49961b69-5f89-485d-ac7f-ca5f2637357c","ref_index":2,"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":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Fisher Larence,Visual Basic.NET: An Introduction to Computer Programming, Kendall Hunt Publishing, USA","work_id":"86224c76-0cff-4f7d-8663-c9d0f1dc3fd0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2020,"title":"Martin Grohe and Pascal Schweitzer,The graph isomorphism problem, Communication ACM,63 (11)(2020), 128–","work_id":"0f860eba-400e-4d08-9161-3bf95d5c109c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"fbea74ca420ab9f471674ce151f899eef3eb38acedbcfe8fe2138442ae597731","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"}