{"paper":{"title":"Edge Boundaries for a Family of Graphs on $\\mathbb{Z}^n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ellen Veomett","submitted_at":"2013-09-12T19:31:01Z","abstract_excerpt":"We consider the family of graphs whose vertex set is $\\mathbb{Z}^n$ where two vertices are connected by an edge when their $\\ell_\\infty$-distance is 1. Towards an edge isoperimetric inequality for this graph, we calculate the edge boundary of any finite set $S \\subset \\mathbb{Z}^n$. This boundary calculation leads to a desire to show that a set with optimal edge boundary has no ``gaps'' in any direction $\\epsilon \\in \\{-1,0,1\\}^n, \\epsilon \\not=0$. We show that one can find a set with optimal edge boundary that does not have gaps in any direction $e_i$ (or $-e_i$) where $e_i$ is the standard b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.3251","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"}