Ju l 2 00 9 CHIP - FIRING GAMES , G - PARKING FUNCTIONS , AND AN EFFICIENT BIJECTIVE PROOF OF THE MATRIX - TREE THEOREM

Kirchhoff's matrix-tree theorem states that the number of spanning trees of a graph G is equal to the value of the determinant of the reduced Laplacian of G. We outline an efficient bijective proof of this theorem, by studying a canonical finite abelian group attached to G whose order is equal to the value of same matrix determinant. More specifically, we show how one can efficiently compute a bijection between the group elements and the spanning trees of the graph. The main ingredient for computing the bijection is an efficient algorithm for finding the unique G-parking function (reduced divisor) in a linear equivalence class defined by a chip-firing game. We also give applications, including a new and completely algebraic algorithm for generating random spanning trees. Other applications include algorithms related to chip-firing games and sandpile group law, as well as certain algorithmic problems about the Riemann-Roch theory on graphs.