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An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V \\rightarrow [1,|V|] $ that has the property that for all $x,y\\in V$ if $(x,y)\\in A$, then $g(x) < g(y)$.\n  We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: \\[\\sum_{x\\in V} e_{_{D}}(x)\\cdot g(x) ~\\geq~ \\frac{1}{2} \\sum_{x\\in V}[e_{_{D}}(x)]^2~.\\] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets"},"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":"1110.3107","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-10-14T02:02:41Z","cross_cats_sorted":[],"title_canon_sha256":"adfc449da58d2ed03d747b080bdf717b275b1fa3a7720d6b3507f6adf6b450b2","abstract_canon_sha256":"fb302d0a1aefe7ec0b449024a14f2f618dd6d32656a94d59036ca5ee1caff2a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:22.814509Z","signature_b64":"1yb2YhjugqgdayruV1FuZnk02+q/2RSd0YvoH7NVeTgfK91hw+9ugUOrRTOGSMJEybZWOavkVSxkOGrhFq7HDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80b12079353084d5b55565b0a9104607e985c777cc0c7f97132a8aa390314321","last_reissued_at":"2026-05-18T02:32:22.814131Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:22.814131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some inequalities for orderings of acyclic digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Imed Zaguia, Thomas Bier","submitted_at":"2011-10-14T02:02:41Z","abstract_excerpt":"Let $D=(V,A)$ be an acyclic digraph. 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