The flow and tension spaces and lattices of signed graphs

This paper is to introduce circuit, bond, flow, and tension spaces and lattices for signed graphs, and to study the relations among these spaces and lattices. The key ingredient is to introduce circuit and bond characteristic vectors so that the desired spaces and lattices can be defined such that their dimensions and ranks match well to that of matroids of signed graphs. The main results can be stated as follows: (1) the classification of minimal directed cuts; (2) the circuit space (lattice) equals flow space (lattice), and the bond space equals the tension space; (3) the bond lattice equals the row lattice of the incidence matrix, and the reduced bond lattice equals the tension lattice; and (4) for unbalanced signed graphs, the module of potentials is isomorphic to the module of tensions if the coefficient ring is 2-torsion free.