# (0, 1)-Matrices, Discrepancy and Preservers

- 2 Downloads

## Abstract

Let *m* and *n* be positive integers, and let *R* = (*r*_{1}, . . . , *r*_{m}) and *S* = (*s*_{1}, . . . , *s*_{n}) be nonnegative integral vectors. Let *A*(*R*,*S*) be the set of all *m* × *n* (0, 1)-matrices with row sum vector *R* and column vector *S*. Let *R* and *S* be nonincreasing, and let *F*(*R*) be the *m* × *n* (0, 1)-matrix, where for each *i*, the *i*th row of *F*(*R*,*S*) consists of *r*_{i} 1’s followed by (*n*−*r*_{i}) 0’s. Let *A* ∈ *A*(*R*,*S*). The discrepancy of A, disc(*A*), is the number of positions in which *F*(*R*) has a 1 and *A* has a 0. In this paper we investigate linear operators mapping *m* × *n* matrices over the binary Boolean semiring to itself that preserve sets related to the discrepancy. In particular, we show that bijective linear preservers of Ferrers matrices are either the identity mapping or, when *m* = *n*, the transpose mapping.

## Keywords

Ferrers matrix row-dense matrix discrepancy linear preserver strong linear preserver## MSC 2010

15A04 15A21 15A86 05B20 05C50## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

The author wishes to thank the referee whose many suggestions improved the presentation.

## References

- [1]
*L. B. Beasley, N. J. Pullman*: Linear operators preserving properties of graphs. Proc. 20th Southeast Conf. on Combinatorics, Graph Theory, and Computing. Congressus Numerantium 70, Utilitas Mathematica Publishing, Winnipeg, 1990, pp. 105–112.Google Scholar - [2]
*A. Berger*: The isomorphic version of Brualdies nestedness is in P, 2017, 7 pages. Available at https://arxiv.org/abs/1602.02536v2.Google Scholar - [3]
*A. Berger, B. Schreck*: The isomorphic version of Brualdi’s and Sanderson’s nestedness. Algorithms (Basel) 10 (2017), Paper No. 74, 12 pages.MathSciNetCrossRefGoogle Scholar - [4]
*R. A. Brualdi, G. J. Sanderson*: Nested species subsets, gaps, and discrepancy. Oecologia 119 (1999), 256–264.CrossRefGoogle Scholar - [5]
*R. A. Brualdi, J. Shen*: Discrepancy of matrices of zeros and ones. Electron. J. Comb. 6 (1999), Research Paper 15, 12 pages.MathSciNetzbMATHGoogle Scholar - [6]
*S. M. Motlaghian, A. Armandnejad, F. J. Hall*: Linear preservers of row-dense matrices. Czech. Math. J. 66 (2016), 847–858.MathSciNetCrossRefGoogle Scholar