{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:3BVUA5VBD5SY6JIHGJ4N6RCORX","short_pith_number":"pith:3BVUA5VB","canonical_record":{"source":{"id":"1809.10342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-27T04:54:15Z","cross_cats_sorted":[],"title_canon_sha256":"98e43d61fec75e5f86e2e17c16623f94378db312faeaed89aa8ce7ac20be7ae8","abstract_canon_sha256":"297f0f9d390f066c594b83b7839b4b8d61fe82520e8097d2b4baf94b5aeb138d"},"schema_version":"1.0"},"canonical_sha256":"d86b4076a11f658f25073278df444e8dd281fe52399fb36e64dd3093fa26176a","source":{"kind":"arxiv","id":"1809.10342","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.10342","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1809.10342v1","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.10342","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"3BVUA5VBD5SY","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3BVUA5VBD5SY6JIH","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3BVUA5VB","created_at":"2026-05-18T12:32:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:3BVUA5VBD5SY6JIHGJ4N6RCORX","target":"record","payload":{"canonical_record":{"source":{"id":"1809.10342","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-27T04:54:15Z","cross_cats_sorted":[],"title_canon_sha256":"98e43d61fec75e5f86e2e17c16623f94378db312faeaed89aa8ce7ac20be7ae8","abstract_canon_sha256":"297f0f9d390f066c594b83b7839b4b8d61fe82520e8097d2b4baf94b5aeb138d"},"schema_version":"1.0"},"canonical_sha256":"d86b4076a11f658f25073278df444e8dd281fe52399fb36e64dd3093fa26176a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:38.574710Z","signature_b64":"EnCa4nyxIM6Ci8gZOoQYiZbpJ11KsCmPDjl21VwskBg2J1mk0pO6ElIUFwgqE48SHqIXFg8E1qboAwQKM24cDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d86b4076a11f658f25073278df444e8dd281fe52399fb36e64dd3093fa26176a","last_reissued_at":"2026-05-18T00:04:38.574180Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:38.574180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.10342","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":"2026-05-18T00:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dqOMaoGvvZe/YOFuoSOKNmeC22eqQoG+T/1QQ3OE+0+7zdNVPQ73cD84zSlpor8STWQ+cOGcKDIgtd7BinbMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T11:02:41.085362Z"},"content_sha256":"0a134841fa1139f5bae6c7aadff53aba75eb367b0c7a35340248ebd427839b75","schema_version":"1.0","event_id":"sha256:0a134841fa1139f5bae6c7aadff53aba75eb367b0c7a35340248ebd427839b75"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:3BVUA5VBD5SY6JIHGJ4N6RCORX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximizing spectral radius and number of spanning trees in bipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ravindra Bapat","submitted_at":"2018-09-27T04:54:15Z","abstract_excerpt":"The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance dis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10342","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":""},"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":"2026-05-18T00:04:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kKU+JpA8BK+iMMihX2jgidbGsRe3AdKAz4VD5SwISLA92r84Ih1pyN7AQPjVU5KhC04F0K/Tip5CQSuR/bYnCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T11:02:41.086096Z"},"content_sha256":"0c9943d8e152677af3e0cfe31fcbfd9af3194829db63d99146ef2bc3fe52e423","schema_version":"1.0","event_id":"sha256:0c9943d8e152677af3e0cfe31fcbfd9af3194829db63d99146ef2bc3fe52e423"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/bundle.json","state_url":"https://pith.science/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/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":"2026-06-10T11:02:41Z","links":{"resolver":"https://pith.science/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX","bundle":"https://pith.science/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/bundle.json","state":"https://pith.science/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3BVUA5VBD5SY6JIHGJ4N6RCORX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:3BVUA5VBD5SY6JIHGJ4N6RCORX","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":"297f0f9d390f066c594b83b7839b4b8d61fe82520e8097d2b4baf94b5aeb138d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-27T04:54:15Z","title_canon_sha256":"98e43d61fec75e5f86e2e17c16623f94378db312faeaed89aa8ce7ac20be7ae8"},"schema_version":"1.0","source":{"id":"1809.10342","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.10342","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"arxiv_version","alias_value":"1809.10342v1","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.10342","created_at":"2026-05-18T00:04:38Z"},{"alias_kind":"pith_short_12","alias_value":"3BVUA5VBD5SY","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_16","alias_value":"3BVUA5VBD5SY6JIH","created_at":"2026-05-18T12:32:02Z"},{"alias_kind":"pith_short_8","alias_value":"3BVUA5VB","created_at":"2026-05-18T12:32:02Z"}],"graph_snapshots":[{"event_id":"sha256:0c9943d8e152677af3e0cfe31fcbfd9af3194829db63d99146ef2bc3fe52e423","target":"graph","created_at":"2026-05-18T00:04:38Z","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":"The problems of maximizing the spectral radius and the number of spanning trees in a class of bipartite graphs with certain degree constraints are considered. In both the problems, the optimal graph is conjectured to be a Ferrers graph. Known results towards the resolution of the conjectures are described. We give yet another proof of a formula due to Ehrenborg and van Willigenburg for the number of spanning trees in a Ferrers graph. The main tool is a result which gives several necessary and sufficient conditions under which the removal of an edge in a graph does not affect the resistance dis","authors_text":"Ravindra Bapat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-27T04:54:15Z","title":"Maximizing spectral radius and number of spanning trees in bipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.10342","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":"sha256:0a134841fa1139f5bae6c7aadff53aba75eb367b0c7a35340248ebd427839b75","target":"record","created_at":"2026-05-18T00:04:38Z","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":"297f0f9d390f066c594b83b7839b4b8d61fe82520e8097d2b4baf94b5aeb138d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-27T04:54:15Z","title_canon_sha256":"98e43d61fec75e5f86e2e17c16623f94378db312faeaed89aa8ce7ac20be7ae8"},"schema_version":"1.0","source":{"id":"1809.10342","kind":"arxiv","version":1}},"canonical_sha256":"d86b4076a11f658f25073278df444e8dd281fe52399fb36e64dd3093fa26176a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d86b4076a11f658f25073278df444e8dd281fe52399fb36e64dd3093fa26176a","first_computed_at":"2026-05-18T00:04:38.574180Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:38.574180Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EnCa4nyxIM6Ci8gZOoQYiZbpJ11KsCmPDjl21VwskBg2J1mk0pO6ElIUFwgqE48SHqIXFg8E1qboAwQKM24cDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:38.574710Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.10342","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0a134841fa1139f5bae6c7aadff53aba75eb367b0c7a35340248ebd427839b75","sha256:0c9943d8e152677af3e0cfe31fcbfd9af3194829db63d99146ef2bc3fe52e423"],"state_sha256":"05ab03c928979edfd74e40f787d30fcb82e1cf28a36496933129f038f7ad4c20"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H1DjsOGyo5chdMQV2JROsu1jFPb2F+KfPTzDSGDcpRHyPBYOBNdtKcphgLdRvPbSzLK8n2hzrkO5D6MkTFcxBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T11:02:41.089961Z","bundle_sha256":"c75f0070171b479bc772101390ca750b1a8bf1952df041f4380bc35b77e0cb0d"}}