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There's a marvelous theorem in graph theory that reduces the count of spanning trees for a graph to a computation of determinant of a naturally-defined matrix (the Laplacian matrix):

I am looking for a book proving the theorem for the following setting: $G$ is a directed graph which allows for multiple edges (which means the Laplacian might contain values other than $-1,0$ off the diagonal). In this case, the minor obtained from removing the $i$th row and column corresponds to the number of spanning tree whose root is $i$.

My usual reference for algebraic graph theory is Godsil and Royle; however, they prove the theorem for a simpler setting (or I don't understand what they do).

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up vote 2 down vote accepted

Dave Perkinson has just what you are looking for:

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