  We study algebraic and homological properties of the ideal of submaximal
minors of a sparse generic symmetric matrix. This ideal is generated by all
$(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper
triangle are distinct variables or zeros and the zeros are only allowed at
off-diagonal places. The surviving off-diagonal entries are encoded as a simple
graph $G$ with $n$ vertices. We prove that the minimal free resolution of this
ideal is obtained from the case without any zeros via a simple pruning
procedure, extending methods of Boocher. This allows us to compute all graded
Betti numbers in terms of $n$ and a single invariant of $G$. Moreover, it turns
out that these ideals are always radical and have Cohen--Macaulay quotients if
and only if $G$ is either connected or has no edges at all. The key input are
some new Gr\"obner basis results with respect to non-diagonal term orders
associated to $G$.
