{"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. For $x\\in V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.3107","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}