  A graph is called Ricci-flat if its Ricci curvatures vanish on all edges,
here the definition of Ricci curvature on graphs was given by Lin-Lu-Yau. The
authors in arXiv:1301.0102 and arXiv:1802.02982 obtained a complete
characterization for all Ricci-flat graphs with girth at least five. In this
paper, we completely determined all Ricci-flat graphs with maximum degree at
most 4.
