{"paper":{"title":"An index formula for simple graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG","math.GN"],"primary_cat":"math.DG","authors_text":"Oliver Knill","submitted_at":"2012-05-02T02:37:04Z","abstract_excerpt":"Gauss-Bonnet for simple graphs G assures that the sum of curvatures K(x) over the vertex set V of G is the Euler characteristic X(G). Poincare-Hopf tells that for any injective function f on V the sum of i(f,x) is X(G). We also know that averaging the indices E[i(f,x)] over all functions gives curvature K(x).\n  We explore here the situation when G is geometric of dimension d: that is if each unit sphere S(x) is geometric of dimension d-1 and that X(S(x))=0 for even d and X(S(x))=2 for odd d. The dimension of G is inductively defined as the average of 1+dim(S(x)) over all S(x) assuming the empt"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0306","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"}