Kirchoff Matrix -Tree Theorem I'm reading a proof of the Kirchoff Matrix -Tree Theorem:

If $G$ is a simple connected graph, $D$ the diagonal matrix with the vertices' degrees and $A$ the adjacency matrix, then in $M = -A+D$ every cofactor is equal to the number of the covering trees of $G$.

I don't understand in the proof the following fact: why all the cofactors of $M$ must be equal ?

Another definition of laplacian matrix is M( or L) = $QQ^t$, where $Q$ is the incidence matrix. By cauchy-binet theorem, one can calculate the determinant of $QQ^t$ by considering $Q$ and $Q^t$. So, if we take a cofactor, then it can contain twigs of a tree and since determinant of matrix containing (#nodes-1) and twigs of a tree is +/- 1. So, multiplying two cofactors each from Q and $Q^t$ will result in only +1. Also, for any other cofactor not containing all edges which are twigs of a tree, its determinant goes to 0. As a result we have determinant of cofactor=#(trees).
You can refer to cauchy-binet theorem on wikipedia.    
