{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:C7QAQ4TQXOYAIM76YMBPMEDOT6","short_pith_number":"pith:C7QAQ4TQ","canonical_record":{"source":{"id":"1501.01741","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T06:31:31Z","cross_cats_sorted":["cs.DM","cs.DS","math.PR","math.SP"],"title_canon_sha256":"2f7943844cfeeccfda61f18546eb23333386d3146be6d54d90a5396d32c611d6","abstract_canon_sha256":"d93973bbfcfcd1d09898e49ffe4add63547f929066981336e5041d622db62fe8"},"schema_version":"1.0"},"canonical_sha256":"17e0087270bbb00433fec302f6106e9fbc4c11165609ec8720ea6f45719087ad","source":{"kind":"arxiv","id":"1501.01741","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01741","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01741v2","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01741","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"pith_short_12","alias_value":"C7QAQ4TQXOYA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"C7QAQ4TQXOYAIM76","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"C7QAQ4TQ","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:C7QAQ4TQXOYAIM76YMBPMEDOT6","target":"record","payload":{"canonical_record":{"source":{"id":"1501.01741","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T06:31:31Z","cross_cats_sorted":["cs.DM","cs.DS","math.PR","math.SP"],"title_canon_sha256":"2f7943844cfeeccfda61f18546eb23333386d3146be6d54d90a5396d32c611d6","abstract_canon_sha256":"d93973bbfcfcd1d09898e49ffe4add63547f929066981336e5041d622db62fe8"},"schema_version":"1.0"},"canonical_sha256":"17e0087270bbb00433fec302f6106e9fbc4c11165609ec8720ea6f45719087ad","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:26:02.373754Z","signature_b64":"4u/Dg8/gPJgyvoENVBdEEhH7seMHu8Vs0JHEPd3XphtSw7lzu6Xr50iCu5FNL+cg8EoQLvt5XVX+INIxmsrZDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"17e0087270bbb00433fec302f6106e9fbc4c11165609ec8720ea6f45719087ad","last_reissued_at":"2026-05-18T02:26:02.373348Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:26:02.373348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.01741","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":"2026-05-18T02:26:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QbS8VcfploakLhPSiggYFxDKa/HfP86qd8zUwR6ffZyQB6HCvE1C+PeATE42Mre8uB+FDS5rcj9/W9EF6YgPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:37:52.367226Z"},"content_sha256":"1a9324e4f83c709c7ef3d5662c32be886be50125d23c8df42d3dbe94d60a6c5f","schema_version":"1.0","event_id":"sha256:1a9324e4f83c709c7ef3d5662c32be886be50125d23c8df42d3dbe94d60a6c5f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:C7QAQ4TQXOYAIM76YMBPMEDOT6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A linear k-fold Cheeger inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.PR","math.SP"],"primary_cat":"math.CO","authors_text":"Franklin Kenter, Mary Radcliffe","submitted_at":"2015-01-08T06:31:31Z","abstract_excerpt":"Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 \\lambda_1 \\le h_G \\le \\sqrt {2 \\lambda_1}$ where $\\lambda_1$ is the minimum nontrivial eigenvalue of the normalized Laplacian matrix.\n  Recent work has generalized the concept of the Cheeger constant when partitioning the vertices of a graph into $k > 2$ parts. While there are several approaches, recent results have shown these higher-order Ch"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01741","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":"2026-05-18T02:26:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z88+76eX/ntQdMXkhgdg6EjF2g32xx5HWX3gv6pMxXFzpfwd57s4KyOTlJsICcsTAjK5eZf2r7OuBHir+SrnDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:37:52.367924Z"},"content_sha256":"72f1c04093cbb1c96bd1fdb23c444587f16a2851461d1ade5c3ed42838ffa2e7","schema_version":"1.0","event_id":"sha256:72f1c04093cbb1c96bd1fdb23c444587f16a2851461d1ade5c3ed42838ffa2e7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/bundle.json","state_url":"https://pith.science/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/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":"2026-05-27T05:37:52Z","links":{"resolver":"https://pith.science/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6","bundle":"https://pith.science/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/bundle.json","state":"https://pith.science/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/C7QAQ4TQXOYAIM76YMBPMEDOT6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:C7QAQ4TQXOYAIM76YMBPMEDOT6","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":"d93973bbfcfcd1d09898e49ffe4add63547f929066981336e5041d622db62fe8","cross_cats_sorted":["cs.DM","cs.DS","math.PR","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T06:31:31Z","title_canon_sha256":"2f7943844cfeeccfda61f18546eb23333386d3146be6d54d90a5396d32c611d6"},"schema_version":"1.0","source":{"id":"1501.01741","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01741","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01741v2","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01741","created_at":"2026-05-18T02:26:02Z"},{"alias_kind":"pith_short_12","alias_value":"C7QAQ4TQXOYA","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"C7QAQ4TQXOYAIM76","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"C7QAQ4TQ","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:72f1c04093cbb1c96bd1fdb23c444587f16a2851461d1ade5c3ed42838ffa2e7","target":"graph","created_at":"2026-05-18T02:26:02Z","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":"Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 \\lambda_1 \\le h_G \\le \\sqrt {2 \\lambda_1}$ where $\\lambda_1$ is the minimum nontrivial eigenvalue of the normalized Laplacian matrix.\n  Recent work has generalized the concept of the Cheeger constant when partitioning the vertices of a graph into $k > 2$ parts. While there are several approaches, recent results have shown these higher-order Ch","authors_text":"Franklin Kenter, Mary Radcliffe","cross_cats":["cs.DM","cs.DS","math.PR","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T06:31:31Z","title":"A linear k-fold Cheeger inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01741","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":"sha256:1a9324e4f83c709c7ef3d5662c32be886be50125d23c8df42d3dbe94d60a6c5f","target":"record","created_at":"2026-05-18T02:26:02Z","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":"d93973bbfcfcd1d09898e49ffe4add63547f929066981336e5041d622db62fe8","cross_cats_sorted":["cs.DM","cs.DS","math.PR","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T06:31:31Z","title_canon_sha256":"2f7943844cfeeccfda61f18546eb23333386d3146be6d54d90a5396d32c611d6"},"schema_version":"1.0","source":{"id":"1501.01741","kind":"arxiv","version":2}},"canonical_sha256":"17e0087270bbb00433fec302f6106e9fbc4c11165609ec8720ea6f45719087ad","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"17e0087270bbb00433fec302f6106e9fbc4c11165609ec8720ea6f45719087ad","first_computed_at":"2026-05-18T02:26:02.373348Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:26:02.373348Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4u/Dg8/gPJgyvoENVBdEEhH7seMHu8Vs0JHEPd3XphtSw7lzu6Xr50iCu5FNL+cg8EoQLvt5XVX+INIxmsrZDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:26:02.373754Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.01741","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1a9324e4f83c709c7ef3d5662c32be886be50125d23c8df42d3dbe94d60a6c5f","sha256:72f1c04093cbb1c96bd1fdb23c444587f16a2851461d1ade5c3ed42838ffa2e7"],"state_sha256":"ecb0b25b4b7b5d9a84c541929611b1a11635d2218b71b6ed845c0a1dcd0b706c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fNlSJDh/ZUkb/nr/mKpiaekODgk62/n8VcdIDcOb03hx5s2dX45bNnSEOESIr2z690FcgkOalRDHd+qlF1yFDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:37:52.371526Z","bundle_sha256":"4fc108c518dab60b8d7f74daf1aacfa331a8f0b124fda0118a3547322b232c09"}}