{"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"}