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This parameter measures how much $P$ deviates from being simple.\n  It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. On the other hand, for fixed $d$, the number of values not taken by the excess degree is finite if $d$ is odd, and the number of even values not taken by the excess degree is finite if $d$ is ev"},"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":"1703.10702","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-30T22:40:46Z","cross_cats_sorted":[],"title_canon_sha256":"feb02b09040cccc38e7c89dd0bd98b2d637234ad87814f9c94487693ec0cc40d","abstract_canon_sha256":"d77b20a710ee42cb5162d3ea771df935cc715e7939d413dd4720cffaf65a1db0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:16.988274Z","signature_b64":"maekvj6whmipl8aLJEObt0lOvpV0FlD0T7IVAbXUIOEHOVbh0u1z7ibPVwYfO8SLk91x/MFHRdkMQAU8TcRZBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6f2098546f9c135205e6cd87551411493aa19558af00a0c9efe3fe16b0b7ff7f","last_reissued_at":"2026-05-18T00:23:16.987613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:16.987613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The excess degree of a polytope","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Yost, Guillermo Pineda-Villavicencio, Julien Ugon","submitted_at":"2017-03-30T22:40:46Z","abstract_excerpt":"We define the excess degree $\\xi(P)$ of a $d$-polytope $P$ as $2f_1-df_0$, where $f_0$ and $f_1$ denote the number of vertices and edges, respectively. This parameter measures how much $P$ deviates from being simple.\n  It turns out that the excess degree of a $d$-polytope does not take every natural number: the smallest possible values are $0$ and $d-2$, and the value $d-1$ only occurs when $d=3$ or 5. 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