{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:FH663OWFXYVLRK3VNA6IAUHJ5Z","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":"eabe890341e5e21f4feeddfca72834073d2c1ba57c2161c2cd3373c6578bef4f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-22T06:40:45Z","title_canon_sha256":"607139623be6d9fd7c3a7e9209ce6ac1119c2f45157b593f24e41539122ba17e"},"schema_version":"1.0","source":{"id":"2606.22904","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.22904","created_at":"2026-06-23T03:14:03Z"},{"alias_kind":"arxiv_version","alias_value":"2606.22904v1","created_at":"2026-06-23T03:14:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.22904","created_at":"2026-06-23T03:14:03Z"},{"alias_kind":"pith_short_12","alias_value":"FH663OWFXYVL","created_at":"2026-06-23T03:14:03Z"},{"alias_kind":"pith_short_16","alias_value":"FH663OWFXYVLRK3V","created_at":"2026-06-23T03:14:03Z"},{"alias_kind":"pith_short_8","alias_value":"FH663OWF","created_at":"2026-06-23T03:14:03Z"}],"graph_snapshots":[{"event_id":"sha256:efe60fcb8d29b8756c0fc5912e780cc66ccb7afd7a5cd5aa4dfd7ade0ada43b0","target":"graph","created_at":"2026-06-23T03:14:03Z","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/2606.22904/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The co-maximal subgroup graph $\\Gamma(G)$ of a finite group $G$ is defined to be a graph with the set of all non-trivial proper subgroups of $G$ as the set of vertices and two distinct vertices $H$ and $K$ are adjacent if and only if $HK=G$. The deleted co-maximal subgroup graph of $G$, denoted by $\\Gamma^*(G)$, is defined as the graph obtained by removing the isolated vertices from $\\Gamma(G)$. In this paper, we prove that for any finite group $G$, $\\Gamma^*(G)$ is connected. Furthermore, we show that $\\Gamma^*(G)$ either contains a cycle or is a star. When $\\Gamma^*(G)$ contains a cycle, its","authors_text":"Angsuman Das, Arnab Mandal, Labani Sarkar","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-22T06:40:45Z","title":"On Connectivity of Comaximal Subgroup Graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.22904","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":"sha256:4dff3e945ede00f29e9371295266fce6640edc149c3abc38d334fd663d47eac6","target":"record","created_at":"2026-06-23T03:14:03Z","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":"eabe890341e5e21f4feeddfca72834073d2c1ba57c2161c2cd3373c6578bef4f","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-06-22T06:40:45Z","title_canon_sha256":"607139623be6d9fd7c3a7e9209ce6ac1119c2f45157b593f24e41539122ba17e"},"schema_version":"1.0","source":{"id":"2606.22904","kind":"arxiv","version":1}},"canonical_sha256":"29fdedbac5be2ab8ab75683c8050e9ee657f88d1d935f71f59dddb0ee57098de","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29fdedbac5be2ab8ab75683c8050e9ee657f88d1d935f71f59dddb0ee57098de","first_computed_at":"2026-06-23T03:14:03.770183Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T03:14:03.770183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oCzj453+pYHH4DGho2GZOPfxfNwEzo1DS3BQ6zrIwtU1lTVuhE5kTsygoPN3OEcgKRn0bucZMGSPIER4caL9Bg==","signature_status":"signed_v1","signed_at":"2026-06-23T03:14:03.770530Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.22904","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4dff3e945ede00f29e9371295266fce6640edc149c3abc38d334fd663d47eac6","sha256:efe60fcb8d29b8756c0fc5912e780cc66ccb7afd7a5cd5aa4dfd7ade0ada43b0"],"state_sha256":"c0ef6da23d9324f4ea62544298c05e21f90c5f091099de4b94fd00862e6ddd30"}