{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:7FF4376NQKILC6WRLEYYXWZQGT","merge_version":"pith-open-graph-merge-v1","event_count":3,"valid_event_count":3,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2cfbb2d26806acb38f93ce895bace4064311355c787c466059d697c41209d92f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T05:39:20Z","title_canon_sha256":"50345caea557d9edcc86483cbf7797e07755b8bf420358a07b139454b546829b"},"schema_version":"1.0","source":{"id":"2605.14402","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.14402","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"2605.14402v1","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.14402","created_at":"2026-05-17T23:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"7FF4376NQKIL","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"7FF4376NQKILC6WR","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"7FF4376N","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:66b96a2ee2617bb592232bc2c2866c9bddb5b900099d77887e56617a73ef41ae","target":"graph","created_at":"2026-05-17T23:39:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"Two families of circulant graphs C_432(R) and C_6750(S) each possess Type-2 isomorphic variants for two values of m."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","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."}],"snapshot_sha256":"f3071905c62dba11867761f3559aecc52ab211d40ca0cf2078e0c6ddd7f30122"},"formal_canon":{"evidence_count":3,"snapshot_sha256":"17ef3e6fac673831b1072beab57ac1e954e3e5f6dd414369a991db8bddc9a2a9"},"paper":{"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}.","authors_text":"Vilfred Kamalappan","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T05:39:20Z","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"},"references":{"count":21,"internal_anchors":1,"resolved_work":21,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"A. Adam,Research problem 2-10, J. Combinatorial Theory,3(1967), 393","work_id":"7a43d6b0-2dfb-4412-99ad-96186a8a7489","year":1967},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"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","year":1982},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"P. J. Davis,Circulant Matrices,Wiley, New York, 1979","work_id":"0f76597d-55e1-4004-87ef-8efe1734b290","year":1979},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"West,Introduction to Graph Theory,2 ed Edi., Pearson Education (Singapore) Pvt","work_id":"cad7d710-39d0-4154-96c0-117798e78ec1","year":2002},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"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","year":1970}],"snapshot_sha256":"3c828d45fe615bc80f52b71e5d470906c8f5bb07a007fc6198bdc9a232f59de2"},"source":{"id":"2605.14402","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-15T01:58:52.160054Z","id":"596cc171-3985-4f1f-b744-f90b945b1610","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"596cc171-3985-4f1f-b744-f90b945b1610"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9834ad25892f5348b2386aa6e6a8168d75799d8e9ffe6c70e0b3cbf1ace63586","target":"record","created_at":"2026-05-17T23:39:07Z","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":"2cfbb2d26806acb38f93ce895bace4064311355c787c466059d697c41209d92f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-14T05:39:20Z","title_canon_sha256":"50345caea557d9edcc86483cbf7797e07755b8bf420358a07b139454b546829b"},"schema_version":"1.0","source":{"id":"2605.14402","kind":"arxiv","version":1}},"canonical_sha256":"f94bcdffcd8290b17ad159318bdb3034cb97953b98f11d5aae0ed9c9e24da1b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f94bcdffcd8290b17ad159318bdb3034cb97953b98f11d5aae0ed9c9e24da1b9","first_computed_at":"2026-05-17T23:39:07.470464Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:07.470464Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KLlr/G5GqcZDX14gjhhR9Vyej3KHghrGkV/vgQf0Hu7d5akX5yf3c9x3qxhg3EaHTy4VtX/unebDgM13CQ01CA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:07.471152Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.14402","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9834ad25892f5348b2386aa6e6a8168d75799d8e9ffe6c70e0b3cbf1ace63586","sha256:66b96a2ee2617bb592232bc2c2866c9bddb5b900099d77887e56617a73ef41ae","sha256:69051fcb4580332e4365589e4855ba9e6abd611b0c378002faccff4e51a3f08d"],"state_sha256":"95aedcf3f05e2b74087f2bb6b2be6e6ddbfbece8a0b3ee906dd3c752009cd167"}