{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:32RPDJXLDKI3Z2RHBL5CNHSXK4","short_pith_number":"pith:32RPDJXL","schema_version":"1.0","canonical_sha256":"dea2f1a6eb1a91bcea270afa269e5757076039d63f000bc4c8a2ef1abc0dc04c","source":{"kind":"arxiv","id":"1412.6075","version":1},"attestation_state":"computed","paper":{"title":"A Generalized Cheeger Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Gary Miller, Ioannis Koutis, Richard Peng","submitted_at":"2014-10-22T15:48:54Z","abstract_excerpt":"The generalized conductance $\\phi(G,H)$ between two graphs $G$ and $H$ on the same vertex set $V$ is defined as the ratio $$\n  \\phi(G,H) = \\min_{S\\subseteq V} \\frac{cap_G(S,\\bar{S})}{ cap_H(S,\\bar{S})}, $$ where $cap_G(S,\\bar{S})$ is the total weight of the edges crossing from $S$ to $\\bar{S}=V-S$. We show that the minimum generalized eigenvalue $\\lambda(L_G,L_H)$ of the pair of Laplacians $L_G$ and $L_H$ satisfies $$\n  \\lambda(L_G,L_H) \\geq \\phi(G,H) \\phi(G)/8, $$ where $\\phi(G)$ is the usual conductance of $G$. A generalized cut that meets this bound can be obtained from the generalized eige"},"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":"1412.6075","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-10-22T15:48:54Z","cross_cats_sorted":[],"title_canon_sha256":"008845344895edd81a3727455e7716d03ebd6c8f6b6e4b9e2f3ce4b900380df9","abstract_canon_sha256":"cfd3d6e925a63b87eb693fade8ebdc03fa13cc9d7d4526a26970bd31f5aa7b78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:55.527656Z","signature_b64":"Dt64dnon6Mgqp9CMog5d7P0mSGpWd1Vd4Tnlr5iEHnffLvydC33jqJRYkW/F0S99d7eCZqYOwzb8Ry0leQuSDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dea2f1a6eb1a91bcea270afa269e5757076039d63f000bc4c8a2ef1abc0dc04c","last_reissued_at":"2026-05-18T02:30:55.527044Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:55.527044Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Generalized Cheeger Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Gary Miller, Ioannis Koutis, Richard Peng","submitted_at":"2014-10-22T15:48:54Z","abstract_excerpt":"The generalized conductance $\\phi(G,H)$ between two graphs $G$ and $H$ on the same vertex set $V$ is defined as the ratio $$\n  \\phi(G,H) = \\min_{S\\subseteq V} \\frac{cap_G(S,\\bar{S})}{ cap_H(S,\\bar{S})}, $$ where $cap_G(S,\\bar{S})$ is the total weight of the edges crossing from $S$ to $\\bar{S}=V-S$. We show that the minimum generalized eigenvalue $\\lambda(L_G,L_H)$ of the pair of Laplacians $L_G$ and $L_H$ satisfies $$\n  \\lambda(L_G,L_H) \\geq \\phi(G,H) \\phi(G)/8, $$ where $\\phi(G)$ is the usual conductance of $G$. A generalized cut that meets this bound can be obtained from the generalized eige"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6075","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":"1412.6075","created_at":"2026-05-18T02:30:55.527157+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.6075v1","created_at":"2026-05-18T02:30:55.527157+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6075","created_at":"2026-05-18T02:30:55.527157+00:00"},{"alias_kind":"pith_short_12","alias_value":"32RPDJXLDKI3","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"32RPDJXLDKI3Z2RH","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"32RPDJXL","created_at":"2026-05-18T12:28:11.866339+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/32RPDJXLDKI3Z2RHBL5CNHSXK4","json":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4.json","graph_json":"https://pith.science/api/pith-number/32RPDJXLDKI3Z2RHBL5CNHSXK4/graph.json","events_json":"https://pith.science/api/pith-number/32RPDJXLDKI3Z2RHBL5CNHSXK4/events.json","paper":"https://pith.science/paper/32RPDJXL"},"agent_actions":{"view_html":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4","download_json":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4.json","view_paper":"https://pith.science/paper/32RPDJXL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.6075&json=true","fetch_graph":"https://pith.science/api/pith-number/32RPDJXLDKI3Z2RHBL5CNHSXK4/graph.json","fetch_events":"https://pith.science/api/pith-number/32RPDJXLDKI3Z2RHBL5CNHSXK4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4/action/storage_attestation","attest_author":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4/action/author_attestation","sign_citation":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4/action/citation_signature","submit_replication":"https://pith.science/pith/32RPDJXLDKI3Z2RHBL5CNHSXK4/action/replication_record"}},"created_at":"2026-05-18T02:30:55.527157+00:00","updated_at":"2026-05-18T02:30:55.527157+00:00"}