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We find an asymptotic expression for the number of multigraphs on the vertex set {1,..., n} with degree sequence given by k, such that every loop has multiplicity in J* and every non-loop edge has multiplicity in J. Equivalently, these are symmetric integer matrices with values J* allowed on the diagonal and J off the diagonal. Our expression holds when the maximum degree K satisfies K = o(M^(1/3)). We prove this result using the swi"},"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":"1303.4218","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-18T11:55:24Z","cross_cats_sorted":[],"title_canon_sha256":"a092d2eecdd88122ca55a7912c0635c58fd5d112b551f74b0831437763b0f64d","abstract_canon_sha256":"ae12d91578bf5d77e38543734a197c36e81128c0d817d7fe2de8a432e5bbf64c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:46.285928Z","signature_b64":"+Xn/x08LF1+4IWe5JtDwYUK23s7RSWJiT5bMh4BrBB6LVPvvvhBPyz2bEm6zwTGVF97ZpvKU+75TV76hBCinAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad4609f75ec1671c3891537e2bee2425dafa2f0e942f469f0dd49e7ca2390ae2","last_reissued_at":"2026-05-18T03:12:46.285216Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:46.285216Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic enumeration of sparse multigraphs with given degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brendan D McKay, Catherine Greenhill","submitted_at":"2013-03-18T11:55:24Z","abstract_excerpt":"Let J and J* be subsets of Z+ such that 0,1\\in J and 0\\in J*. 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