Irreducible matrices with reducible principal submatrices Original Research Article

Let A be a (0, 1)-matrix of order n greater-or-equal, slanted 3 and let si0(A), i = 1, …, n, be the number of the off diagonal 0ʹs in row and column i of A. We prove that if A is irreducible, and if all its principal submatrices of order (n − 1) are reducible, then si0(A) greater-or-equal, slanted n − 1; i = 1, …, n. This establishes the validity of a conjecture by B. Schwarz concerning strongly connected graphs and their primal subgraphs.