Kirchhoff's theorem relies on the notion of the Laplacian matrix of a graph that is equal to the difference between the graph's degree matrix (a diagonal matrix with vertex degrees on the diagonals) and its adjacency matrix (a $(0,1)$-matrix with $1$'s at places corresponding to entries where the vertices are adjacent and $0$'s otherwise).
It is known that for a given connected graph $G$ with $n$ labeled vertices, let $λ_1,λ_2,...,λ_{n−1}$ be the non-zero eigenvalues of its Laplacian matrix. Then the number of spanning trees of $G$ is $t(G)=\frac{1}{n} \lambda_1\lambda_2\cdots\lambda_{n-1}\,$
Now I want to extend it to the steiner tree. A $T$-Steiner tree is a subgraph of $G$ that is a tree and that spans $T$. When $T=V(G)$, a $T$-Steiner tree is a spanning tree. My question is that what is the connection between general $T$-Steiner tree and the Laplacian matrix? So far I didn't find any known results about this extension. Your comments and discussion are welcome.