# Hall exponents of matrices, tournaments and their line digraphs

• Volume: 61, Issue: 2, page 461-481
• ISSN: 0011-4642

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## Abstract

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Let $A$ be a square $\left(0,1\right)$-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).

## How to cite

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Brualdi, Richard A., and Kiernan, Kathleen P.. "Hall exponents of matrices, tournaments and their line digraphs." Czechoslovak Mathematical Journal 61.2 (2011): 461-481. <http://eudml.org/doc/196346>.

@article{Brualdi2011,
abstract = {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).},
author = {Brualdi, Richard A., Kiernan, Kathleen P.},
journal = {Czechoslovak Mathematical Journal},
keywords = {Hall matrix; Hall exponent; irreducible; primitive; tournament (matrix); line digraph; Hall matrix; Hall exponent; irreducible; primitive; tournament (matrix); line digraph; -matrix; permanent; adjacency matrix},
language = {eng},
number = {2},
pages = {461-481},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Hall exponents of matrices, tournaments and their line digraphs},
url = {http://eudml.org/doc/196346},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Brualdi, Richard A.
AU - Kiernan, Kathleen P.
TI - Hall exponents of matrices, tournaments and their line digraphs
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 2
SP - 461
EP - 481
AB - 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).
LA - eng
KW - Hall matrix; Hall exponent; irreducible; primitive; tournament (matrix); line digraph; Hall matrix; Hall exponent; irreducible; primitive; tournament (matrix); line digraph; -matrix; permanent; adjacency matrix
UR - http://eudml.org/doc/196346
ER -

## References

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6. Hemminger, R. L., Beineke, L. W., Line graphs and line digraphs, Selected Topics in Graph Theory Academic Press, New York, 271-305 (1978). (1978) Zbl0434.05056
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8. Liu, B., Zhou, Bo, Estimates on strict Hall exponents, Australas. J. Comb. 19 (1999), 129-135. (1999) MR1695868
9. Liu, B., Zhou, Bo, 10.1080/03081089908818611, Linear Multilinear Algebra 46 (1999), 165-175. (1999) MR1708586DOI10.1080/03081089908818611
10. Moon, J. W., Pullman, N. J., 10.1016/S0021-9800(67)80009-7, J. Comb. Theory 3 (1967), 1-9. (1967) Zbl0166.00901MR0213264DOI10.1016/S0021-9800(67)80009-7
11. Schwarz, Š., The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Math. J. 23(98) (1973), 151-163. (1973) Zbl0261.20057MR0316612
12. Tan, S., Liu, B., Zhang, D., Extreme tournaments with given primitive exponents, Australas. J. Comb. 28 (2003), 81-91. (2003) MR1998863

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