  We introduce some determinantal ideals of the generalized Laplacian matrix
associated to a digraph G, that we call critical ideals of G. Critical ideals
generalize the critical group and the characteristic polynomials of the
adjacency and Laplacian matrices of a digraph. The main results of this article
are the determination of some minimal generator sets and the reduced Grobner
basis for the critical ideals of the complete graphs, the cycles and the paths.
Also, we establish a bound between the number of trivial critical ideals and
the stability and clique numbers of a graph.
