{"paper":{"title":"Families of $2$-weights of some particular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Agnese Baldisserri, Elena Rubei","submitted_at":"2016-05-03T15:25:42Z","abstract_excerpt":"Let ${\\cal G}=(G,w) $ be a positive-weighted graph, that is a graph $G$ endowed with a function $w$ from the edge set of $G$ to the set of positive real numbers; for any distinct vertices $i,j $, we define $D_{i,j}({\\cal G})$ to be the weight of the path in $G$ joining $i$ and $j$ with minimum weight. In this paper we fix a particular class of graphs and we give a criterion to establish whether, given a family of positive real numbers $\\{D_I\\}_{I \\in { \\{1,...., n\\} \\choose 2}}$, there exists a positive-weighted graph ${\\cal G} =(G,w) $ in the class we have fixed, with vertex set equal to $\\{1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.00946","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"}