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We prove that in both cases the number of rooted spanning forests can be represented in the form $f_{G}(n)=p\\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the parity of $n$. Finally, we find an asymptotic formula for $f_{G}(n)$ through the Mahler measure of the associated "},"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":"1907.02635","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-07-04T03:03:47Z","cross_cats_sorted":[],"title_canon_sha256":"3bc20ce4043a0d88e7af6450ab41c94a181f594bdcc4b78d3cbaa6f948248f0a","abstract_canon_sha256":"cc7741ee19a54f99d1f4113c124ce4d6f86339f8cfae625f05398d47f5263c92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:23.686741Z","signature_b64":"2L9w190mLD5EojOK4SPiQx1hWk+YKH7zbLJUR+Fbx4Ee/qpHRnLbq5liZZSLKrkBC5C1wBs0+9Larf6LtHKmAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14e1d14dba6123eae6d43421694c60c7a356ddd249428a9a719881d0971a092e","last_reissued_at":"2026-05-17T23:41:23.686040Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:23.686040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The number of rooted forests in circulant graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"I.A. Mednykh, L.A. Grunwald","submitted_at":"2019-07-04T03:03:47Z","abstract_excerpt":"In this paper, we develop a new method to produce explicit formulas for the number $f_{G}(n)$ of rooted spanning forests in the circulant graphs $ G=C_{n}(s_1,s_2,\\ldots,s_k)$ and $ G=C_{2n}(s_1,s_2,\\ldots,s_k,n).$ These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form $f_{G}(n)=p\\,a(n)^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending on the parity of $n$. 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