  In this paper, we compare Ollivier Ricci curvature and Bakry-\'Emery
curvature notions on combinatorial graphs and discuss connections to various
types of Ricci flatness. We show that non-negativity of Ollivier Ricci
curvature implies non-negativity of Bakry-\'Emery curvature under
triangle-freeness and an additional in-degree condition. We also provide
examples that both conditions of this result are necessary. We investigate
relations to graph products and show that Ricci flatness is preserved under all
natural products. While non-negativity of both curvatures are preserved under
Cartesian products, we show that in the case of strong products, non-negativity
of Ollivier Ricci curvature is only preserved for horizontal and vertical
edges. We also prove that all distance-regular graphs of girth $4$ attain their
maximal possible curvature values.
