{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:37S5NETWF4KJFQKBMEDT3C4HH2","short_pith_number":"pith:37S5NETW","canonical_record":{"source":{"id":"2606.27328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-25T17:41:10Z","cross_cats_sorted":[],"title_canon_sha256":"72710de92e9dcd1260e071d46f368de74ca26714cdf1d78d0d89a4a88129111f","abstract_canon_sha256":"e7dd9575aefc554569b41069e40575571690282036488220b5b8ad6e4105e336"},"schema_version":"1.0"},"canonical_sha256":"dfe5d692762f1492c14161073d8b873eb0e732ed921cc29056c7c31a9a92ae64","source":{"kind":"arxiv","id":"2606.27328","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27328","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27328v1","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27328","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_12","alias_value":"37S5NETWF4KJ","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_16","alias_value":"37S5NETWF4KJFQKB","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_8","alias_value":"37S5NETW","created_at":"2026-06-26T01:16:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:37S5NETWF4KJFQKBMEDT3C4HH2","target":"record","payload":{"canonical_record":{"source":{"id":"2606.27328","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-25T17:41:10Z","cross_cats_sorted":[],"title_canon_sha256":"72710de92e9dcd1260e071d46f368de74ca26714cdf1d78d0d89a4a88129111f","abstract_canon_sha256":"e7dd9575aefc554569b41069e40575571690282036488220b5b8ad6e4105e336"},"schema_version":"1.0"},"canonical_sha256":"dfe5d692762f1492c14161073d8b873eb0e732ed921cc29056c7c31a9a92ae64","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:16:18.831825Z","signature_b64":"dv+agV/N1O/oD/ptm8JCjU2QYoxL1FW2TNJPp88wtosiEi9mKI8GVmUVp5ddvm6mwnauRubnLV+XwPl690CsBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfe5d692762f1492c14161073d8b873eb0e732ed921cc29056c7c31a9a92ae64","last_reissued_at":"2026-06-26T01:16:18.831416Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:16:18.831416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.27328","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lR3lfiz1V/LEPyPr+mz46G9PRvVGUTVVe7FNjND3qVhzGTJWo7WRKulWqMDPmM5Vj6fzFA24q/VxUzKsVVL0CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:22:40.380896Z"},"content_sha256":"6e2026501bb937470ee2cbbd1c5efed76c5998f0fc6678f914425c63902783c8","schema_version":"1.0","event_id":"sha256:6e2026501bb937470ee2cbbd1c5efed76c5998f0fc6678f914425c63902783c8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:37S5NETWF4KJFQKBMEDT3C4HH2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the homology groups of clique complexes of strongly regular graphs","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Mutasim Mim, Sebastian M. Cioab\\u{a}","submitted_at":"2026-06-25T17:41:10Z","abstract_excerpt":"In this paper, we study the first homology groups of clique complexes of strongly regular graphs over arbitrary fields and prove that most of these graphs have trivial first clique homology groups. Using Neumaier's classification of strongly regular graphs with smallest integral eigenvalue, we show that a non-vanishing first homology group may occur only in a short collection of cases: the Petersen graph, the Shrikhande graph, the complete bipartite graphs, the conference graphs on at most $255$ vertices, the lattice graphs, and the exceptional families $E_m$ in Neumaier's classification of st"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27328","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27328/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9whdLQQMfskwnM1JfBfPxGoNHOa237jJdHWoAsqNmzRgxGiO+kD1ykWdSAXLKbitmIiGu0SFx4pze4PUFycRAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T18:22:40.381479Z"},"content_sha256":"b109816f5f06253ec99f0abc44d7f054e8cad10d81aacde53a67cba4d77237fe","schema_version":"1.0","event_id":"sha256:b109816f5f06253ec99f0abc44d7f054e8cad10d81aacde53a67cba4d77237fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/37S5NETWF4KJFQKBMEDT3C4HH2/bundle.json","state_url":"https://pith.science/pith/37S5NETWF4KJFQKBMEDT3C4HH2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/37S5NETWF4KJFQKBMEDT3C4HH2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-30T18:22:40Z","links":{"resolver":"https://pith.science/pith/37S5NETWF4KJFQKBMEDT3C4HH2","bundle":"https://pith.science/pith/37S5NETWF4KJFQKBMEDT3C4HH2/bundle.json","state":"https://pith.science/pith/37S5NETWF4KJFQKBMEDT3C4HH2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/37S5NETWF4KJFQKBMEDT3C4HH2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:37S5NETWF4KJFQKBMEDT3C4HH2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e7dd9575aefc554569b41069e40575571690282036488220b5b8ad6e4105e336","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-25T17:41:10Z","title_canon_sha256":"72710de92e9dcd1260e071d46f368de74ca26714cdf1d78d0d89a4a88129111f"},"schema_version":"1.0","source":{"id":"2606.27328","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27328","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27328v1","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27328","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_12","alias_value":"37S5NETWF4KJ","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_16","alias_value":"37S5NETWF4KJFQKB","created_at":"2026-06-26T01:16:18Z"},{"alias_kind":"pith_short_8","alias_value":"37S5NETW","created_at":"2026-06-26T01:16:18Z"}],"graph_snapshots":[{"event_id":"sha256:b109816f5f06253ec99f0abc44d7f054e8cad10d81aacde53a67cba4d77237fe","target":"graph","created_at":"2026-06-26T01:16:18Z","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":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.27328/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the first homology groups of clique complexes of strongly regular graphs over arbitrary fields and prove that most of these graphs have trivial first clique homology groups. Using Neumaier's classification of strongly regular graphs with smallest integral eigenvalue, we show that a non-vanishing first homology group may occur only in a short collection of cases: the Petersen graph, the Shrikhande graph, the complete bipartite graphs, the conference graphs on at most $255$ vertices, the lattice graphs, and the exceptional families $E_m$ in Neumaier's classification of st","authors_text":"Mutasim Mim, Sebastian M. Cioab\\u{a}","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-25T17:41:10Z","title":"On the homology groups of clique complexes of strongly regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27328","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e2026501bb937470ee2cbbd1c5efed76c5998f0fc6678f914425c63902783c8","target":"record","created_at":"2026-06-26T01:16:18Z","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":"e7dd9575aefc554569b41069e40575571690282036488220b5b8ad6e4105e336","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-25T17:41:10Z","title_canon_sha256":"72710de92e9dcd1260e071d46f368de74ca26714cdf1d78d0d89a4a88129111f"},"schema_version":"1.0","source":{"id":"2606.27328","kind":"arxiv","version":1}},"canonical_sha256":"dfe5d692762f1492c14161073d8b873eb0e732ed921cc29056c7c31a9a92ae64","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dfe5d692762f1492c14161073d8b873eb0e732ed921cc29056c7c31a9a92ae64","first_computed_at":"2026-06-26T01:16:18.831416Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:16:18.831416Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dv+agV/N1O/oD/ptm8JCjU2QYoxL1FW2TNJPp88wtosiEi9mKI8GVmUVp5ddvm6mwnauRubnLV+XwPl690CsBQ==","signature_status":"signed_v1","signed_at":"2026-06-26T01:16:18.831825Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.27328","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e2026501bb937470ee2cbbd1c5efed76c5998f0fc6678f914425c63902783c8","sha256:b109816f5f06253ec99f0abc44d7f054e8cad10d81aacde53a67cba4d77237fe"],"state_sha256":"6f5dd86305523f9cd443343d13def13a5ae262044bbf120c670be791a8a60765"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gJy5Ayuc2PdshYgK74JX3Q77yY4N135Bdj7/qR+JNYxizgKWbWGfyC35LVfMJqPCaFSOJCZV9ICgYZfbaS8ICQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T18:22:40.383685Z","bundle_sha256":"86ecd4039597b461a84f243f42fe3b90d227be9e8d88ba47f7153df7445e7af1"}}