In a graphic matroid $M(G)$, circuits correspond directly to cycles in the original graph $G$. This means that any property that can be defined for both a circuit in $M(G)$ and for a cycle in $G$ holds for both.
However, as matroids are more broad than just graphs, it can be expected that there are properties of cycles that:
- can be defined for both graphs and matroids
- that hold for cycles in graphs (and hold for circuits in graphic matroids)
- but do not hold for matroids in general.
I am looking for simple, educational examples of such properties.
You can easily construct properties that matroid circuits lack, but they usually involve description using "vertices" or "sequence of edges", and cannot be translated into matroids at all.
You can define orientation of a matroid using the axioms on circuits in the link. All graphic matroids can be oriented, that is you can always oriented (sign) the circuits of a graphic. However, in general you cannot do this with the circuits an arbitrary matroid.