Entropy estimates and volume rigidity for graphs with positive Bakry-Émery curvature

We establish a volume bound for weighted graphs with positive Bakry–Émery curvature through entropy estimates for the heat flow and characterize the equality case. As a consequence, if a connected unweighted graph $G=(V,E)$ satisfies the Bakry–Émery curvature–dimension condition $CD(2,\mathrm{\infty })$ with respect to the unnormalized Laplacian, then $$\mathrm{\mid }V\mathrm{\mid }\le 2^d,$$ where $d$ denotes the maximum degree. Equality holds if and only if $G$ is a hypercube. We also establish an analogous volume rigidity result for the normalized Laplacian.