Tutte polynomials and G-parking functions

In this paper, we give a new expression for the Tutte polynomial of a general connected graph G in terms of statistics of G-parking functions. In particular, given a G-parking function f, let cb G ( f ) be the number of critical-bridge vertices of f and denote w G ( f ) = | E ( G ) | − | V ( G ) | − ∑ v ∈ V ( G ) f ( v ) . We prove that T G ( x , y ) = ∑ f ∈ P G x cb G ( f ) y w G ( f ) , where P G is the set of G-parking functions. Our proof avoids any use of spanning trees and is independent of bijections between the set of G-parking functions and the set of spanning trees.