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A similar result for the weighted complexity of weighted graphs was found using a determinant function [\\textit{J. Combin. Theory Ser. B}, 89 (2003), pp. 17--26]. In this paper, we consider the determinant function of two variables and discover a condition that the weighted complexity of a weighted graph is a partial derivative of the determinant funct"},"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":"0705.2284","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.CO","submitted_at":"2007-05-16T06:51:20Z","cross_cats_sorted":[],"title_canon_sha256":"db08808ccede553dae31370fb46ea63f152e1070ade1bf86b58f015e46d758a4","abstract_canon_sha256":"07f72aa1f034810c3ad771d7576bfb373df81fae33ffd0f485c5bdf85338b328"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:14.574372Z","signature_b64":"sasMeIHdezwcQngCzMhWc1xN8h5lXDXBPwjYznUzG7FiQN3UYlRTsyrcEqsEqthzFR0Qh49XXVz/j8QQoEe6Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c62fc018e8698bc007a2d8121a810846c7b26c7cf21d563dfb50e37fa7167593","last_reissued_at":"2026-05-18T04:38:14.573832Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:14.573832Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The weighted complexity and the determinant functions of graphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dongseok Kim, Jaeun Lee, Young Soo Kwon","submitted_at":"2007-05-16T06:51:20Z","abstract_excerpt":"The complexity of a graph can be obtained as a derivative of a variation of the zeta function or a partial derivative of its generalized characteristic polynomial evaluated at a point [\\textit{J. 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