Edge-regular graphs with non-negative curvature have polynomial growth

A long-standing conjecture in the emerging discrete Bakry-Émery theory asserts that bounded-degree graphs satisfying $\mathrm{C}\mathrm{D}(0,\mathrm{\infty })$ have polynomial growth. In the present paper, we prove this conjecture for all edge-regular graphs, and even obtain a volume doubling estimate with a constant that depends only on the degree. This is made possible thanks to the discovery of a surprising self-improvement phenomenon, which seems of independent interest: any edge-regular graph satisfying $\mathrm{C}\mathrm{D}(\kappa ,\mathrm{\infty })$ for some $\kappa \in \mathbb{R}$ must in fact satisfy $\mathrm{C}\mathrm{D}(\kappa ,n)$ for some explicit, universal and optimal dimension parameter $n$.