Ideals of submaximal minors of sparse symmetric matrices

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 $G$.