{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:2ISFBJT3IXDX6QMESYGRYNEBCS","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":"d03372662dc8c9fde2dfb219e58f509339d450466243203cf7dc94974ae40b55","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-23T16:05:33Z","title_canon_sha256":"73cac46b173e073cb90b4356c7c6eeab0bf237c5ea2eb69785ef44e22ef94e51"},"schema_version":"1.0","source":{"id":"1210.6277","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.6277","created_at":"2026-05-18T03:26:42Z"},{"alias_kind":"arxiv_version","alias_value":"1210.6277v2","created_at":"2026-05-18T03:26:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.6277","created_at":"2026-05-18T03:26:42Z"},{"alias_kind":"pith_short_12","alias_value":"2ISFBJT3IXDX","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"2ISFBJT3IXDX6QME","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"2ISFBJT3","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:d5d8b914a2ea8fcc7936e9c6c1e22df94ea118733736a0a1407a6b9fc86faa2e","target":"graph","created_at":"2026-05-18T03:26:42Z","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":"Bounds are proved for the connective constant \\mu\\ of an infinite, connected, \\Delta-regular graph G. The main result is that \\mu\\ \\ge \\sqrt{\\Delta-1} if G is vertex-transitive and simple. This inequality is proved subject to weaker conditions under which it is sharp.","authors_text":"Geoffrey R. Grimmett, Zhongyang Li","cross_cats":["math-ph","math.MP","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-23T16:05:33Z","title":"Bounds on connective constants of regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6277","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":"sha256:8206d4c6d3a805d146d88eb2e5151a21297d6a75e38321a0c04925aa1377ed1f","target":"record","created_at":"2026-05-18T03:26:42Z","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":"d03372662dc8c9fde2dfb219e58f509339d450466243203cf7dc94974ae40b55","cross_cats_sorted":["math-ph","math.MP","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-10-23T16:05:33Z","title_canon_sha256":"73cac46b173e073cb90b4356c7c6eeab0bf237c5ea2eb69785ef44e22ef94e51"},"schema_version":"1.0","source":{"id":"1210.6277","kind":"arxiv","version":2}},"canonical_sha256":"d22450a67b45c77f4184960d1c34811487d6bddf4b2e1abc4dc7fb5aa2dd7134","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d22450a67b45c77f4184960d1c34811487d6bddf4b2e1abc4dc7fb5aa2dd7134","first_computed_at":"2026-05-18T03:26:42.759957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:42.759957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aAqn7sfyAI/nkygI1yFi1e15BcxQD0c65apKT5Nr3H2M8W5TaKMsJicCJd1embI2AN9XUqlr0fmFVWnrZQm+Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:42.760631Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.6277","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8206d4c6d3a805d146d88eb2e5151a21297d6a75e38321a0c04925aa1377ed1f","sha256:d5d8b914a2ea8fcc7936e9c6c1e22df94ea118733736a0a1407a6b9fc86faa2e"],"state_sha256":"76c2178f4ea3db38c8ea73ea5a7b7884851e881ece2ccb42f1586edb09fb660c"}