The **independence complex** $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.


Now, if $G$ is a **[well-covered graph](https://en.wikipedia.org/wiki/Well-covered_graph)** (where all maximal independent sets have the same cardinality $\alpha$) then $I(G)$ is obviously the uniform matroid $U\_{n,\alpha}$.



> 
> Can the graphs $G$ for which $I(G)$ is a matroid be characterized?
> 
> 
> 



Which graphs generate a matroidal independence complex?
A (finite) simplicial complex is the independent set complex of a graph $G$ if and only if its minimal nonfaces have two elements. If it is also a matroid complex, if follows that the circuits of the matroid have two elements. Thus $G$ is a disjoint union of complete graphs, not just a single complete graph.


