Assume a $d$-regular graph $G$ has a perfect matching. I am wondering is the character function of the perfect matchings anything special in the matching polytope $P$ that is the convex hull of $$\{\mathbb{1}_M\in \mathbb{R}^{|E(G)|}| \text{ $M$ is a matching of $G$}\},$$ where $\mathbb{1}_M$ if the character function of $M$, i.e., $\mathbb{1}_M(e)=1$ if $e\in M$ and 0 otherwise.
One theorem says the vertices of $P$ are formed by the character function of matchings.