Geodesic
Hall exponents of matrices, tournaments and their line digraphs
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 461-481
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Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that $A^k$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).
Let $A$ be a square $(0,1)$-matrix. Then $A$ is a Hall matrix provided it has a nonzero permanent. The Hall exponent of $A$ is the smallest positive integer $k$, if such exists, such that $A^k$ is a Hall matrix. The Hall exponent has received considerable attention, and we both review and expand on some of its properties. Viewing $A$ as the adjacency matrix of a digraph, we prove several properties of the Hall exponents of line digraphs with some emphasis on line digraphs of tournament (matrices).
DOI : 10.1007/s10587-011-0066-2
Classification : 05C20, 15A15, 15B34
Keywords: Hall matrix; Hall exponent; irreducible; primitive; tournament (matrix); line digraph 
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Brualdi, Richard A.; Kiernan, Kathleen P. Hall exponents of matrices, tournaments  and their line digraphs. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 2, pp. 461-481. doi: 10.1007/s10587-011-0066-2

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