I am stuyding my combinatorics syllabus and came across two claims, that are said to be generalisations of the Matrix Tree Theorem:
G = (V,E) is a complete graph without loops. U is a subset of V. The matrix Q^U(G) is the matrix retreived from the Laplacian matrix L(G) by removing the rows and columns that correspond to the vertices in U.
- Let e = (v,w) be an element of E.Then, the number of spanning trees of G that contain e is equal to det Q^{v,w} (G)
- Let T = (V_T, E_T) with V_T a subset of (but not equal to) V, be a subtree of G. The number of spanning trees of G that contain T is equal to det Q^{V_T}(G)
The proof is presented as an exercise for the reader, but I cannot figure it out. By drawing random graphs, I understand that the claims, but I do not get yet why they are true.
Could somebody help me out? Thanks!