Algebraic constructions of the minimal forbidden digraphs of strong sign nonsingular matrices Original Research Article

A square real matrix A is called a strong sign nonsingular matrix (S2NS matrix) if all the matrices with the same sign pattern as A are nonsingular and all the inverses of these matrices have the same sign pattern. S2NS digraphs are digraphs associated with those S2NS matrices with negative main diagonals. In this paper, we define the associated linear system of equations L(D) (over the finite field F2) for each digraph D, and then define an undirected graph G(L(D)) representing certain relations between the equations of L(D). We obtain algebraic criteria to recognize the minimal forbidden configurations of S2NS digraphs in terms of the solvability of the linear system L(D) and some of its subsystems and the connectedness of the undirected graph G(L(D)). These algebraic criteria together with a conjunction operation of digraphs can be used to construct infinitely many new minimal forbidden configurations.