{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:OKEJWJOAC4U3LWXY77EWBOUWOI","short_pith_number":"pith:OKEJWJOA","schema_version":"1.0","canonical_sha256":"72889b25c01729b5daf8ffc960ba96720a1585fe7ad8b8d74cef891ce7954256","source":{"kind":"arxiv","id":"1111.0965","version":1},"attestation_state":"computed","paper":{"title":"Many Sparse Cuts via Higher Eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Anand Louis, Prasad Raghavendra, Prasad Tetali, Santosh Vempala","submitted_at":"2011-11-03T19:57:26Z","abstract_excerpt":"Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \\[ \\phi(S) \\defeq \\frac{w(S,\\bar{S})}{\\min \\set{w(S), w(\\bar{S})}} \\leq 2\\sqrt{\\lambda_2} \\] where $w$ is the total edge weight of a subset or a cut and $\\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \\in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that \\[ \\max_i \\phi(S_i) \\leq C \\sqrt{\\lambda_{k} \\log k} "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.0965","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-11-03T19:57:26Z","cross_cats_sorted":[],"title_canon_sha256":"25adff48c238c4516ff008d8d5e6e097f3a211449d9af5b0431d3970a0f22746","abstract_canon_sha256":"40db63f8c272a50f7530b050bab2a52932ef18c0f347a8c1dca9ec5d127b56f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:34.927987Z","signature_b64":"AQAeDyINZAJiiEz3IOC5MQVTUx2YZKgvc/jChRx+YPhCoBdfL5c4ZHszOoTsSh587t4LPN1cLNYqNp9LEBp0Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72889b25c01729b5daf8ffc960ba96720a1585fe7ad8b8d74cef891ce7954256","last_reissued_at":"2026-05-18T02:21:34.927560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:34.927560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Many Sparse Cuts via Higher Eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Anand Louis, Prasad Raghavendra, Prasad Tetali, Santosh Vempala","submitted_at":"2011-11-03T19:57:26Z","abstract_excerpt":"Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \\[ \\phi(S) \\defeq \\frac{w(S,\\bar{S})}{\\min \\set{w(S), w(\\bar{S})}} \\leq 2\\sqrt{\\lambda_2} \\] where $w$ is the total edge weight of a subset or a cut and $\\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \\in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that \\[ \\max_i \\phi(S_i) \\leq C \\sqrt{\\lambda_{k} \\log k} "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.0965","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1111.0965","created_at":"2026-05-18T02:21:34.927630+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.0965v1","created_at":"2026-05-18T02:21:34.927630+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.0965","created_at":"2026-05-18T02:21:34.927630+00:00"},{"alias_kind":"pith_short_12","alias_value":"OKEJWJOAC4U3","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_16","alias_value":"OKEJWJOAC4U3LWXY","created_at":"2026-05-18T12:26:37.096874+00:00"},{"alias_kind":"pith_short_8","alias_value":"OKEJWJOA","created_at":"2026-05-18T12:26:37.096874+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI","json":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI.json","graph_json":"https://pith.science/api/pith-number/OKEJWJOAC4U3LWXY77EWBOUWOI/graph.json","events_json":"https://pith.science/api/pith-number/OKEJWJOAC4U3LWXY77EWBOUWOI/events.json","paper":"https://pith.science/paper/OKEJWJOA"},"agent_actions":{"view_html":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI","download_json":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI.json","view_paper":"https://pith.science/paper/OKEJWJOA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.0965&json=true","fetch_graph":"https://pith.science/api/pith-number/OKEJWJOAC4U3LWXY77EWBOUWOI/graph.json","fetch_events":"https://pith.science/api/pith-number/OKEJWJOAC4U3LWXY77EWBOUWOI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI/action/storage_attestation","attest_author":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI/action/author_attestation","sign_citation":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI/action/citation_signature","submit_replication":"https://pith.science/pith/OKEJWJOAC4U3LWXY77EWBOUWOI/action/replication_record"}},"created_at":"2026-05-18T02:21:34.927630+00:00","updated_at":"2026-05-18T02:21:34.927630+00:00"}