{"paper":{"title":"Conductance and Eigenvalue","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"cs.DM","authors_text":"Girish Varma","submitted_at":"2010-09-09T12:39:41Z","abstract_excerpt":"We show the following. \\begin{theorem} Let $M$ be an finite-state ergodic time-reversible Markov chain with transition matrix $P$ and conductance $\\phi$. Let $\\lambda \\in (0,1)$ be an eigenvalue of $P$. Then, $$\\phi^2 + \\lambda^2 \\leq 1$$ \\end{theorem} This strengthens the well-known~\\cite{HLW,Dod84, AM85, Alo86, JS89} inequality $\\lambda \\leq 1- \\phi^2/2$. We obtain our result by a slight variation in the proof method in \\cite{JS89, HLW}; the same method was used earlier in \\cite{RS06} to obtain the same inequality for random walks on regular undirected graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.1756","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"}