Kirchhoff's theorem for directed multigraphs

Statement

Kirchhoff's theorem can be modified to give the number of oriented spanning trees in directed multigraphs. The matrix Q is constructed as follows:

• The entry q_i,j for distinct i and j equals −m, where m is the number of edges from i to j;
• The entry q_i,i equals the indegree of i minus the number of loops at i.

The number of oriented spanning trees rooted at a vertex i is the determinant of the matrix gotten by removing the ith row and column of Q