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Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between  two maximum separated nodes on an $M \\times N$ rectangular network of resistors with resistors $r$ and $s$ in the two spatial directions. Our results is $ \\frac{1}{s}R_{M\\times N}(r,s)= c(\\rho)\\, \\l"},"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":"1005.1434","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2010-05-10T01:40:22Z","cross_cats_sorted":[],"title_canon_sha256":"4259eaed012e48098bf0419f20a45b479e1201aecba55d74af8a4fe4730f040a","abstract_canon_sha256":"1f2667abd09ca8e073b682a039a159a7b4ef073995e665c276852f7eb6e08ba0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:53.477215Z","signature_b64":"kYxrr5XTcD93uUHRD60wQPrLe8QKUB6V/ea1jYFs80XGfkzeNjbUr1FuSaHhxBm2ION3+5YbEDxbatP5jidsCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7fff499238e1391ad81055acb6f577e37f6ff51661581c83c6e49d1b4d3fecc","last_reissued_at":"2026-05-18T04:19:53.476558Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:53.476558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic expansion for the resistance between two maximum separated nodes on a $M \\times N$ resistor network","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Ming-Chang Huang, N. Sh. Izmailian","submitted_at":"2010-05-10T01:40:22Z","abstract_excerpt":"We analyze the exact formulae for the resistance between two arbitrary notes in a rectangular network of resistors under free, periodic and cylindrical boundary conditions obtained by Wu [J. Phys. A 37, 6653 (2004)]. Based on such expression, we then apply the algorithm of Ivashkevich, Izmailian and Hu [J. Phys. A 35, 5543 (2002)] to derive the exact asymptotic expansions of the resistance between  two maximum separated nodes on an $M \\times N$ rectangular network of resistors with resistors $r$ and $s$ in the two spatial directions. 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