TITLE: Graph-theoretic interpretation of determinant?
QUESTION [5 upvotes]: The permanent can be interpreted as the number of perfect matchings in bipartite graphs.

Is there a similar graph-theoretic interpretation of the determinant?

REPLY [13 votes]: 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.