Let $G$ be a connected undirected graph. Then the number $ST(G)$ of spanning trees in $G$ equals the following specific value of the Tutte polynomial of $G$: $ST(G)=T_G(1,1)$.
On the other hand, Kirchhoff's matrix tree theorem expresses $ST(G)$ as a principal minor of Laplacian of $G$.
The question is wether the whole Tutte polynomial or at least some other values can be expressed as determinants. If in general they can't, I wonder what is behind this phenomenon.