{"paper":{"title":"The graph bottleneck identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.NI","math.MG"],"primary_cat":"math.CO","authors_text":"Pavel Chebotarev","submitted_at":"2010-03-20T03:56:55Z","abstract_excerpt":"A matrix $S=(s_{ij})\\in{\\mathbb R}^{n\\times n}$ is said to determine a \\emph{transitional measure} for a digraph $G$ on $n$ vertices if for all $i,j,k\\in\\{1,\\...,n\\},$ the \\emph{transition inequality} $s_{ij} s_{jk}\\le s_{ik} s_{jj}$ holds and reduces to the equality (called the \\emph{graph bottleneck identity}) if and only if every path in $G$ from $i$ to $k$ contains $j$. We show that every positive transitional measure produces a distance by means of a logarithmic transformation. Moreover, the resulting distance $d(\\cdot,\\cdot)$ is \\emph{graph-geodetic}, that is, $d(i,j)+d(j,k)=d(i,k)$ hold"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.3904","kind":"arxiv","version":4},"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"}