Algebraic constructions of the minimal forbidden digraphs of strong sign nonsingular matrices

Abstract

A square real matrix A is called a strong sign nonsingular matrix ( S 2 NS 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. S 2 NS digraphs are digraphs associated with those S 2 NS matrices with negative main diagonals. In this paper, we define the associated linear system of equations L(D) (over the finite field F 2 ) 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 S 2 NS 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.