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We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$.\n  We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \\in \\{4,5\\}$. Moreover, we des"},"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":"2511.21144","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-11-26T07:56:46Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"d8b71ee27e660a1bab1eac61b6af46e06eef190d193f07bb8506fff774eb3883","abstract_canon_sha256":"682beb96b92c322557e2cf9676f2f72ca2875048cc4bd7574aed470cde337a8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:26.703777Z","signature_b64":"ERGy8OSeJRzjh3u6pJFMzrDiWZdmitgtc88Z2x7lj5OVztkuVhjisVUJ7eSp8jKfjSMYW2ym9seMwaz9DOXABQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e3e6acc81c038f4a116366bf36af23010dea3b4e235cba4b5bc5b5e1d6dc091","last_reissued_at":"2026-06-29T01:14:26.703299Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:26.703299Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the order-diameter ratio of girth-diameter cages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Jan Goedgebeur, Jorik Jooken, Stijn Cambie, Tibo Van den Eede","submitted_at":"2025-11-26T07:56:46Z","abstract_excerpt":"For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$.\n  We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \\in \\{4,5\\}$. 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