Graphs with nonnegative curvature outside a finite subset, harmonic functions, and number of ends

Abstract We study graphs with nonnegative Bakry–Émery curvature or Ollivier curvature outside a finite subset. For such a graph, via introducing the discrete Gromov–Hausdorff convergence, we prove that the space of bounded harmonic functions is finite dimensional and, as a corollary, the number of nonparabolic ends is finite.