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This paper determines $\\zeta_h$ and $\\eta_h$ for the hypercube $Q_n$ and the star graph $S_n$, and $\\eta_3$ for the bubble-sort network $B_n$."},"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":"1511.04113","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-12T22:21:19Z","cross_cats_sorted":[],"title_canon_sha256":"f66401274d8bad0479d58c32211593c53858ba3ba9f192563070b23406b432ad","abstract_canon_sha256":"314f2dcd68e0d52b8d76e2a7dc909bc960153a2deaa6388ff203f29fab2d409a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:26:59.653891Z","signature_b64":"ZxwQjt2oQWNIYlQnr17qWHVU0tSF0y7voYC7sqkdvS+0YYnWCv/+9L5TKwnvMsF0D0mDxtMSZbVYZ344qX6bBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f2a2847924b46919187ca27adf2a800565b60cc674d3a68a3cf4f962e3443b4a","last_reissued_at":"2026-05-18T01:26:59.653186Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:26:59.653186Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Embedded connectivity of recursive networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun-Ming Xu, Qi-Qi Dong, Xiang-jun Li, Zheng Yan","submitted_at":"2015-11-12T22:21:19Z","abstract_excerpt":"Let $G_n$ be an $n$-dimensional recursive network. 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