# 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

## Access Full Article

top## Abstract

top## How to cite

topBrualdi, 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

top- Aigner, M., 10.1007/BF01110285, Math. Z. 102 (1967), 56-61. (1967) MR0216981DOI10.1007/BF01110285
- Brualdi, R. A., Kiernan, K. P., Landau's and Rado's theorems and partial tournaments, Electron. J. Comb. 16 (2009). (2009) Zbl1159.05014MR2475542
- Brualdi, R. A., Li, Q., The interchange graph of tournaments with the same score vector, In: Progress in Graph Theory (Waterloo, Ont. 1982) Academic Press Toronto, ON (1984), 129-151. (1984) Zbl0553.05039MR0776798
- Brualdi, R. A., Liu, B., Hall exponents of Boolean matrices, Czechoslovak Math. J. 40(115) (1990), 659-670. (1990) Zbl0737.05028MR1084901
- Brualdi, R. A., Ryser, H. J., Combinatorial Matrix Theory, Cambridge Univ. Press Cambridge (1991). (1991) Zbl0746.05002MR1130611
- 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
- Huang, Y., Liu, B., 10.1080/03081080902843554, Linear Multilinear Algebra 58 (2010), 699-710. (2010) Zbl1201.15014MR2722753DOI10.1080/03081080902843554
- Liu, B., Zhou, Bo, Estimates on strict Hall exponents, Australas. J. Comb. 19 (1999), 129-135. (1999) MR1695868
- Liu, B., Zhou, Bo, 10.1080/03081089908818611, Linear Multilinear Algebra 46 (1999), 165-175. (1999) MR1708586DOI10.1080/03081089908818611
- 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
- Schwarz, Š., The semigroup of fully indecomposable relations and Hall relations, Czechoslovak Math. J. 23(98) (1973), 151-163. (1973) Zbl0261.20057MR0316612
- Tan, S., Liu, B., Zhang, D., Extreme tournaments with given primitive exponents, Australas. J. Comb. 28 (2003), 81-91. (2003) MR1998863

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.