Graph-theoretic interpretation of determinant? The permanent can be interpreted as the number of perfect matchings in bipartite graphs. > Is there a similar graph-theoretic interpretation of the determinant?

I'm aware of a few. There is the Lindström-Gessel-Viennot lemma, and there is also the matrix-tree theorem. If $A$ is the adjacency matrix of a finite graph $G$ then $\frac{1}{\det(I - At)}$ describes a kind of "zeta function" of $G$. I describe some of how this works in this blog post.

You may also be interested in Kuperberg's An exploration of the permanent-determinant method.