Discrete curvature has recently been used in graph machine learning to improve performance, understand message-passing and assess structural differences between graphs. Despite these advancements, the theoretical properties of discrete curvature measures, such as their representational power and their relationship to graph features is yet to be fully explored. This paper studies Ollivier--Ricci curvature on graphs, providing both a discussion and empirical analysis of its expressivity, i.e. the ability to distinguish non-isomorphic graphs.