Parking functions on nonsingular M-matrices

In this paper, let Δ be a nonsingular M-matrix. A generalization of G-parking functions, which is called Δ-parking functions, is studied. An explicit characterization for Δ-parking functions is given. It is shown that Δ-parking functions can be obtained by a simple way from recurrent configurations on the nonsingular M-matrix Δ. It is proved that the number of Δ-parking functions is equal to the determinant of Δ.