{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:QHI4GFPQPBIMEFPLY337IL6XZE","short_pith_number":"pith:QHI4GFPQ","canonical_record":{"source":{"id":"1805.01056","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-02T23:34:58Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3c38a455a391ec42b209abb2450a10e613cd6be9884d30cde77076e716697da9","abstract_canon_sha256":"6071523615da830bc5738c4a1f471624df53a1482c2d1bd7a6fd2772c6b4fa2f"},"schema_version":"1.0"},"canonical_sha256":"81d1c315f07850c215ebc6f7f42fd7c936f0de49d67787e8d64256488513c112","source":{"kind":"arxiv","id":"1805.01056","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01056","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01056v2","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01056","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"QHI4GFPQPBIM","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QHI4GFPQPBIMEFPL","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QHI4GFPQ","created_at":"2026-05-18T12:32:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:QHI4GFPQPBIMEFPLY337IL6XZE","target":"record","payload":{"canonical_record":{"source":{"id":"1805.01056","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-02T23:34:58Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"3c38a455a391ec42b209abb2450a10e613cd6be9884d30cde77076e716697da9","abstract_canon_sha256":"6071523615da830bc5738c4a1f471624df53a1482c2d1bd7a6fd2772c6b4fa2f"},"schema_version":"1.0"},"canonical_sha256":"81d1c315f07850c215ebc6f7f42fd7c936f0de49d67787e8d64256488513c112","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:52:18.389614Z","signature_b64":"1NWyDQvCTN4hLp49TKwKz8J3LVrgsZjPvOgy92WJS9ix31NReUymPapxzyHe3pfhBt53c2aE4ltngp0MuL9xBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81d1c315f07850c215ebc6f7f42fd7c936f0de49d67787e8d64256488513c112","last_reissued_at":"2026-05-17T23:52:18.389034Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:52:18.389034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.01056","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":"2026-05-17T23:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2u+ExCgc0T+XPaAND3q6i4lTE3iNYN8r4mh5wRifzLsCZQiFypGkjip6rJBZuy6a7HfJAcBudtEsmmDZQ5wsDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:12:28.140235Z"},"content_sha256":"ba75fc158c9b045bcedfb2df6548186e1c50f781ad5e07bbde844c8358c2d6d5","schema_version":"1.0","event_id":"sha256:ba75fc158c9b045bcedfb2df6548186e1c50f781ad5e07bbde844c8358c2d6d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:QHI4GFPQPBIMEFPLY337IL6XZE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A spectral version of the Moore problem for bipartite regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Hiroshi Nozaki, Jack H. Koolen, Sebastian M. Cioab\\u{a}","submitted_at":"2018-05-02T23:34:58Z","abstract_excerpt":"Let $b(k,\\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\\theta$. In this paper, we obtain a general upper bound for $b(k,\\theta)$ for any $0\\leq \\theta< 2\\sqrt{k-1}$. Our bound gives the exact value of $b(k,\\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\\theta$, diameter $d$ and girth $g$ such that $g\\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\\geq 15$, we prove that there are no bi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01056","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":"2026-05-17T23:52:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"20UiZGQ2whNRZtpwusQscWdB8B8FVlen66W5/m12Yp1xlMGm1kaJWn46WfApjzxy5FtArqn0DKRqF5JrCIkWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:12:28.140594Z"},"content_sha256":"7e9435e301135b607561b7a5137678d3708bb347f4c93e52a636c7b295279f79","schema_version":"1.0","event_id":"sha256:7e9435e301135b607561b7a5137678d3708bb347f4c93e52a636c7b295279f79"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QHI4GFPQPBIMEFPLY337IL6XZE/bundle.json","state_url":"https://pith.science/pith/QHI4GFPQPBIMEFPLY337IL6XZE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QHI4GFPQPBIMEFPLY337IL6XZE/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":"2026-05-28T10:12:28Z","links":{"resolver":"https://pith.science/pith/QHI4GFPQPBIMEFPLY337IL6XZE","bundle":"https://pith.science/pith/QHI4GFPQPBIMEFPLY337IL6XZE/bundle.json","state":"https://pith.science/pith/QHI4GFPQPBIMEFPLY337IL6XZE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QHI4GFPQPBIMEFPLY337IL6XZE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:QHI4GFPQPBIMEFPLY337IL6XZE","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":"6071523615da830bc5738c4a1f471624df53a1482c2d1bd7a6fd2772c6b4fa2f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-02T23:34:58Z","title_canon_sha256":"3c38a455a391ec42b209abb2450a10e613cd6be9884d30cde77076e716697da9"},"schema_version":"1.0","source":{"id":"1805.01056","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.01056","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"arxiv_version","alias_value":"1805.01056v2","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.01056","created_at":"2026-05-17T23:52:18Z"},{"alias_kind":"pith_short_12","alias_value":"QHI4GFPQPBIM","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_16","alias_value":"QHI4GFPQPBIMEFPL","created_at":"2026-05-18T12:32:46Z"},{"alias_kind":"pith_short_8","alias_value":"QHI4GFPQ","created_at":"2026-05-18T12:32:46Z"}],"graph_snapshots":[{"event_id":"sha256:7e9435e301135b607561b7a5137678d3708bb347f4c93e52a636c7b295279f79","target":"graph","created_at":"2026-05-17T23:52:18Z","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"},"paper":{"abstract_excerpt":"Let $b(k,\\theta)$ be the maximum order of a connected bipartite $k$-regular graph whose second largest eigenvalue is at most $\\theta$. In this paper, we obtain a general upper bound for $b(k,\\theta)$ for any $0\\leq \\theta< 2\\sqrt{k-1}$. Our bound gives the exact value of $b(k,\\theta)$ whenever there exists a bipartite distance-regular graph of degree $k$, second largest eigenvalue $\\theta$, diameter $d$ and girth $g$ such that $g\\geq 2d-2$. For certain values of $d$, there are infinitely many such graphs of various valencies $k$. However, for $d=11$ or $d\\geq 15$, we prove that there are no bi","authors_text":"Hiroshi Nozaki, Jack H. Koolen, Sebastian M. Cioab\\u{a}","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-02T23:34:58Z","title":"A spectral version of the Moore problem for bipartite regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.01056","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:ba75fc158c9b045bcedfb2df6548186e1c50f781ad5e07bbde844c8358c2d6d5","target":"record","created_at":"2026-05-17T23:52:18Z","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":"6071523615da830bc5738c4a1f471624df53a1482c2d1bd7a6fd2772c6b4fa2f","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-02T23:34:58Z","title_canon_sha256":"3c38a455a391ec42b209abb2450a10e613cd6be9884d30cde77076e716697da9"},"schema_version":"1.0","source":{"id":"1805.01056","kind":"arxiv","version":2}},"canonical_sha256":"81d1c315f07850c215ebc6f7f42fd7c936f0de49d67787e8d64256488513c112","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81d1c315f07850c215ebc6f7f42fd7c936f0de49d67787e8d64256488513c112","first_computed_at":"2026-05-17T23:52:18.389034Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:52:18.389034Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1NWyDQvCTN4hLp49TKwKz8J3LVrgsZjPvOgy92WJS9ix31NReUymPapxzyHe3pfhBt53c2aE4ltngp0MuL9xBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:52:18.389614Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.01056","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba75fc158c9b045bcedfb2df6548186e1c50f781ad5e07bbde844c8358c2d6d5","sha256:7e9435e301135b607561b7a5137678d3708bb347f4c93e52a636c7b295279f79"],"state_sha256":"81bd7f1616951f3222c584c42bbad22a8143ce242fd218a0425a160a498570d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BzPOpujdpGpisS7n3biiGt+8ldUW+JJzX5laSDWltTVT8xLKH1a7D8WKeVItcCbZ4x6+Ta8gs9+a1gJQbuqAAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T10:12:28.142608Z","bundle_sha256":"3e9d1aaf4ec3a77f0ca8a090058ccd71cdd0b7c290da17389890530314e181d7"}}