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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."},"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":"1009.1756","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"cs.DM","submitted_at":"2010-09-09T12:39:41Z","cross_cats_sorted":[],"title_canon_sha256":"7bda3658b9d342bc67d75f0dcb339e11b17d10c5caf61459a9c2d06fc2ba7a97","abstract_canon_sha256":"33017bfa4440c076863d574a6c919b41762606fac08208be53d2d10517cf12f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:06.341611Z","signature_b64":"3iwOITCpAmC/xan0V5//0h76QoUJSxfVN+ReMQXOCvBUaeaUj6o3/99zMMUlLHOOdUeERO+WUmpL+T9yOnBzAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"75e149c72b0e46945e655a42c9ef9c5294f2a729019ab8342a92b918f6b74ddd","last_reissued_at":"2026-05-18T04:41:06.341211Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:06.341211Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\lambda \\leq 1- \\phi^2/2$. 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