(0, 1)-Matrices, Discrepancy and Preservers

  • LeRoy B. BeasleyEmail author


Let m and n be positive integers, and let R = (r1, . . . , rm) and S = (s1, . . . , sn) 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 ith row of F(R,S) consists of ri 1’s followed by (nri) 0’s. Let AA(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.


Ferrers matrix row-dense matrix discrepancy linear preserver strong linear preserver 

MSC 2010

15A04 15A21 15A86 05B20 05C50 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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


  1. [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. [2]
    A. Berger: The isomorphic version of Brualdies nestedness is in P, 2017, 7 pages. Available at Scholar
  3. [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. [4]
    R. A. Brualdi, G. J. Sanderson: Nested species subsets, gaps, and discrepancy. Oecologia 119 (1999), 256–264.CrossRefGoogle Scholar
  5. [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. [6]
    S. M. Motlaghian, A. Armandnejad, F. J. Hall: Linear preservers of row-dense matrices. Czech. Math. J. 66 (2016), 847–858.MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUtah State UniversityLoganUSA

Personalised recommendations