{"paper":{"title":"Nonnegative Bakry--\\'Emery Curvature on Bounded-Degree Graphs Implies Volume Doubling and Poincar\\'e Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.DG","authors_text":"Qi Guo, Xueping Huang, Yi C. Huang","submitted_at":"2026-07-17T00:13:49Z","abstract_excerpt":"We prove that every connected simple graph of bounded degree satisfying the classical dimension-free Bakry--\\'Emery condition $\\mathrm{CD}(0,\\infty)$ for the unnormalised Laplacian is volume doubling and supports, at all integer graph scales, a scale-invariant $L^2$-Poincar\\'e inequality with dilation two, with constants depending only on the maximum degree. This settles the polynomial-growth conjecture of Cushing, Liu, and Peyerimhoff in a stronger form. The main novelty is a dimension-free adaptation of the graph-theoretic modified nonlinear heat-flow method introduced by M\\\"unch and extende"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15522","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.15522/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"}