Matrix Tree Theorems

Abstract A simple proof of a directed graph generalization of the Matrix Tree Theorem, sometimes called Maxwell's rule or Kirchhoff's rule, is given. It is based on the idea A. Renyi used to prove Cayley's tree counting formula. The theorem counts rooted arborescences (analogs of forests) in a directed graph with the determinant of a submatrix in a special adjacency matrix. In the proof we show two n - k degree homogeneous polynomials in n variables are equal by applying induction to those terms lacking one variable. An application to a well-known identity and related theorems are given.