{"paper":{"title":"Metrization of weighted graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Matti Vuorinen, Oleksiy Dovgoshey, Olli Martio","submitted_at":"2011-05-31T06:25:40Z","abstract_excerpt":"We find a set of necessary and sufficient conditions under which the weight $w:E\\to\\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\\times V\\to\\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\\mathfrak M_w$ with respect to the partial ordering $d_1 \\leqslant d_2$ if and only if $d_1(u,v) \\leqslant d_2(u,v)$ for all $u,v\\in V$. It is shown that every nonvoid poset $(\\mathfrak M_w,\\leqslant)$ contains the least element $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.6167","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"}