{"paper":{"title":"Gauss-Bonnet for multi-linear valuations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GN"],"primary_cat":"cs.DM","authors_text":"Oliver Knill","submitted_at":"2016-01-18T14:36:46Z","abstract_excerpt":"We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constructing higher order Dehn-Sommerville valuations which vanish for all d-graphs without boundary. An example of a quadratic valuation is the Wu characteristic w(G) which sums (-1)^(dim(x)+dim(y)) over all intersecting pairs of complete subgraphs x,y of a G. More generally, an intersection number w(A,B) sums (-1)^(dim(x)+dim(y)) over pairs x,y, where x is in A and y is in B and x,y intersect. w(G) is a quadratic Euler ch"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04533","kind":"arxiv","version":1},"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"}