{"paper":{"title":"Graph topologies induced by edge lengths","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Agelos Georgakopoulos","submitted_at":"2009-03-10T16:37:15Z","abstract_excerpt":"Let $G$ be a graph each edge $e$ of which is given a length $\\ell(e)$. This naturally induces a distance $d_\\ell(x,y)$ between any two vertices $x,y$, and we let $\\ell-TOP$ denote the completion of the corresponding metric space. It turns out that several well studied topologies on infinite graphs are special cases of $\\ell-TOP$. Moreover, it seems that $\\ell-TOP$ is the right setting for studying various problems. The aim of this paper is to introduce $\\ell-TOP$, providing basic facts, motivating examples and open problems, and indicate possible applications."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1744","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0903.1744/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}