{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UBY5U5EB5FOV3PHRXOVIFQXRKQ","short_pith_number":"pith:UBY5U5EB","canonical_record":{"source":{"id":"1505.08094","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-29T15:51:23Z","cross_cats_sorted":[],"title_canon_sha256":"3c95518ec0a916390c31cf3b9a04851a94d367b0a797b6e140dfba3fb926b833","abstract_canon_sha256":"3c89f1e38c3bbc7e9f71cfb009fff877d1c2c19d0955852edc6d3b68104d8da3"},"schema_version":"1.0"},"canonical_sha256":"a071da7481e95d5dbcf1bbaa82c2f154033055dae2609a9c5c752b3a4f4ddf2c","source":{"kind":"arxiv","id":"1505.08094","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.08094","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"arxiv_version","alias_value":"1505.08094v2","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.08094","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"pith_short_12","alias_value":"UBY5U5EB5FOV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UBY5U5EB5FOV3PHR","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UBY5U5EB","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UBY5U5EB5FOV3PHRXOVIFQXRKQ","target":"record","payload":{"canonical_record":{"source":{"id":"1505.08094","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-29T15:51:23Z","cross_cats_sorted":[],"title_canon_sha256":"3c95518ec0a916390c31cf3b9a04851a94d367b0a797b6e140dfba3fb926b833","abstract_canon_sha256":"3c89f1e38c3bbc7e9f71cfb009fff877d1c2c19d0955852edc6d3b68104d8da3"},"schema_version":"1.0"},"canonical_sha256":"a071da7481e95d5dbcf1bbaa82c2f154033055dae2609a9c5c752b3a4f4ddf2c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:34.447543Z","signature_b64":"fkyBcD99YSNS8vw7EFC5zZMlYSLfZFXvszjwDW5tqksLecMTTj2QVjlh0pMkUYu6p+ufvrL8Y7+6XI4CIU2lBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a071da7481e95d5dbcf1bbaa82c2f154033055dae2609a9c5c752b3a4f4ddf2c","last_reissued_at":"2026-05-18T01:59:34.447039Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:34.447039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1505.08094","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:59:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UzJyGz2VVhkaAD8sOh3zk3RumfXg/evhYW+OCDci8BRr3da0xycq9IctsDHR14NYrjInb/bWSLbWmtZA4nfoDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T17:10:51.360815Z"},"content_sha256":"ed2b59ffc740ac395c773b36223b5e09824c2c938634f85e80a75b80a5e9a748","schema_version":"1.0","event_id":"sha256:ed2b59ffc740ac395c773b36223b5e09824c2c938634f85e80a75b80a5e9a748"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UBY5U5EB5FOV3PHRXOVIFQXRKQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Toroidality and projective-planarity of intersection graphs of subgroups of finite groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"P. Devi, R. Rajkumar","submitted_at":"2015-05-29T15:51:23Z","abstract_excerpt":"Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\\mathscr{I}(G)$ are adjacent if and only if the corresponding subgroups having a non-trivial intersection in $G$. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or total"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.08094","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"2026-05-18T01:59:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"A0Sf0D3A+RtpXLJDxiw6Iu4rWw1Q0AM4UEQZo+WARgrSLifIT5TPDzMGMgDDtF8LOe38hbnr8RzDmX2mild3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T17:10:51.361162Z"},"content_sha256":"8b3c7770199eeba7fb895eccb3a1d4435f398f5c3e4de64d1558b90e5b5b495b","schema_version":"1.0","event_id":"sha256:8b3c7770199eeba7fb895eccb3a1d4435f398f5c3e4de64d1558b90e5b5b495b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/bundle.json","state_url":"https://pith.science/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/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":"2026-06-22T17:10:51Z","links":{"resolver":"https://pith.science/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ","bundle":"https://pith.science/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/bundle.json","state":"https://pith.science/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UBY5U5EB5FOV3PHRXOVIFQXRKQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UBY5U5EB5FOV3PHRXOVIFQXRKQ","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":"3c89f1e38c3bbc7e9f71cfb009fff877d1c2c19d0955852edc6d3b68104d8da3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-29T15:51:23Z","title_canon_sha256":"3c95518ec0a916390c31cf3b9a04851a94d367b0a797b6e140dfba3fb926b833"},"schema_version":"1.0","source":{"id":"1505.08094","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.08094","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"arxiv_version","alias_value":"1505.08094v2","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.08094","created_at":"2026-05-18T01:59:34Z"},{"alias_kind":"pith_short_12","alias_value":"UBY5U5EB5FOV","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UBY5U5EB5FOV3PHR","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UBY5U5EB","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:8b3c7770199eeba7fb895eccb3a1d4435f398f5c3e4de64d1558b90e5b5b495b","target":"graph","created_at":"2026-05-18T01:59:34Z","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"},"paper":{"abstract_excerpt":"Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\\mathscr{I}(G)$ are adjacent if and only if the corresponding subgroups having a non-trivial intersection in $G$. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or total","authors_text":"P. Devi, R. Rajkumar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-29T15:51:23Z","title":"Toroidality and projective-planarity of intersection graphs of subgroups of finite groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.08094","kind":"arxiv","version":2},"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:ed2b59ffc740ac395c773b36223b5e09824c2c938634f85e80a75b80a5e9a748","target":"record","created_at":"2026-05-18T01:59:34Z","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":"3c89f1e38c3bbc7e9f71cfb009fff877d1c2c19d0955852edc6d3b68104d8da3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-05-29T15:51:23Z","title_canon_sha256":"3c95518ec0a916390c31cf3b9a04851a94d367b0a797b6e140dfba3fb926b833"},"schema_version":"1.0","source":{"id":"1505.08094","kind":"arxiv","version":2}},"canonical_sha256":"a071da7481e95d5dbcf1bbaa82c2f154033055dae2609a9c5c752b3a4f4ddf2c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a071da7481e95d5dbcf1bbaa82c2f154033055dae2609a9c5c752b3a4f4ddf2c","first_computed_at":"2026-05-18T01:59:34.447039Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:59:34.447039Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fkyBcD99YSNS8vw7EFC5zZMlYSLfZFXvszjwDW5tqksLecMTTj2QVjlh0pMkUYu6p+ufvrL8Y7+6XI4CIU2lBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:59:34.447543Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.08094","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed2b59ffc740ac395c773b36223b5e09824c2c938634f85e80a75b80a5e9a748","sha256:8b3c7770199eeba7fb895eccb3a1d4435f398f5c3e4de64d1558b90e5b5b495b"],"state_sha256":"55a900baa37db93a47484a2f218f3b5a73a385367cde211103902efc274d724a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1rP9xBf8hDRjKd/Z1lBjAC9G+eKg58NzQRBS2iMjSjAYM5LaZBuXIGP8PG/Fh3nFLZk1Yz+vgZa+fHgORa5FBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T17:10:51.363132Z","bundle_sha256":"24c998697e384c9ca90679c9d739344226c39d5d9e172f1b00daa883c91dce09"}}