I've got a symmetric matrix $A$ whose entries are in $\{0,-1,1\}$, with the diagonal entries all equal to $1$. I'm interested in finding a combinatorial description of the entries of the inverse of $A$ (when it exists).

Now the snag is that this looks tantalizingly close to a *matrix-tree type* problem but not quite, because in a matrix-tree type problem each diagonal entry is equal to the negative of the sum of the off-diagonal entries in its row. I could not find a good way to reduce my problem to this case because nothing prevents the off-diagonal sum from being zero in my case, so scaling seems to be out of the question and additive changes would probably destroy any structure.

Is there some other way to relate my problem to matrix-tree?