  Lin-Lu-Yau introduced an interesting notion of Ricci curvature for graphs and
obtained a complete characterization for all Ricci-flat graphs with girth at
least five [1]. In this paper, we propose a concrete approach to construct an
infinite family of distinct Ricci-flat graphs of girth four with edge-disjoint
4-cycles and completely characterize all Ricci-flat graphs of girth four with
vertex-disjoint 4-cycles.
