{"paper":{"title":"Bakry-\\'Emery curvature functions of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CO","authors_text":"David Cushing, Norbert Peyerimhoff, Shiping Liu","submitted_at":"2016-06-05T11:30:11Z","abstract_excerpt":"We study the Bakry-\\'Emery curvature function $\\mathcal{K}_{G,x}:(0,\\infty]\\to \\mathbb{R}$ of a vertex $x$ in a locally finite graph $G$ systematically. Here $\\mathcal{K}_{G,x}(\\mathcal{N})$ is defined as the optimal curvature lower bound $\\mathcal{K}$ in the Bakry-\\'Emery curvature-dimension inequality $CD(\\mathcal{K},\\mathcal{N})$ that $x$ satisfies. We prove the curvature functions of the Cartesian product of two graphs $G_1,G_2$ equal an abstract product of curvature functions of $G_1,G_2$. We relate the curvature functions of $G$ with various spectral properties of (weighted) graphs const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01496","kind":"arxiv","version":2},"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"}