Irreducible matrices with reducible principal submatrices |
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Authors: | David London |
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Affiliation: | Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel |
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Abstract: | Let A be a (0, 1)-matrix of order n 3 and let si0(A), i = 1, …, n, be the number of the off diagonal 0's 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) n − 1; i = 1, …, n. This establishes the validity of a conjecture by B. Schwarz concerning strongly connected graphs and their primal subgraphs. |
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Keywords: | Irreducible matrix Principal submatrix Strongly connected digraph Primal subgraph |
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