Hall exponents of matrices, tournaments and their line digraphs

Richard A. Brualdi; Kathleen P. Kiernan

Czechoslovak Mathematical Journal (2011)

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

Abstract

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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).

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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  4. Brualdi, R. A., Liu, B., Hall exponents of Boolean matrices, Czechoslovak Math. J. 40(115) (1990), 659-670. (1990) Zbl0737.05028MR1084901
  5. Brualdi, R. A., Ryser, H. J., Combinatorial Matrix Theory, Cambridge Univ. Press Cambridge (1991). (1991) Zbl0746.05002MR1130611
  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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