Matrices over finite fields and their Kirchhoff graphs

Given a stoichiometric matrix A with integer entries, we seek to visualize the information by a reaction network N, whose edges are labeled by the columns of A, such that all column dependencies are realized as cycles in N. Moreover, the vector edges of N are assigned integer multiplicities so that every vertex of N corresponds to a vector in the row space of A. A finite such N is called a Kirchhoff graph. We establish the existence of nontrivial Kirchhoff graphs over finite fields, but the general problem over the integers is still open.