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 cbG(f) be the number of critical-bridge vertices of f and denote wG(f)=|E(G)|−|V(G)|−∑v∈V(G)f(v). We prove that TG(x,y)=∑f∈PGxcbG(f)ywG(f), where PG 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.