  For a graph $G$ with $p$ vertices the closed convex cone
$\mathbb{S}^p_{\succeq0}(G)$ consists of all real positive semidefinite
$p\times p$ matrices with zeros in the off-diagonal entries corresponding to
nonedges of $G$. The extremal rays of this cone and their associated ranks have
applications to matrix completion problems, maximum likelihood estimation in
Gaussian graphical models in statistics, and Gauss elimination for sparse
matrices. For a graph $G$ without $K_5$ minors, we show that the normal vectors
to the facets of the $(\pm1)$-cut polytope of $G$ specify the off-diagonal
entries of extremal matrices in $\mathbb{S}^p_{\succeq0}(G)$. We also prove
that the constant term of the linear equation of each facet-supporting
hyperplane is the rank of its corresponding extremal matrix in
$\mathbb{S}^p_{\succeq0}(G)$. Furthermore, we show that if $G$ is
series-parallel then this gives a complete characterization of all possible
extremal ranks of $\mathbb{S}^p_{\succeq0}(G)$, consequently solving the
sparsity order problem for series-parallel graphs.
