# Article

 Title: $(0,1)$-matrices, discrepancy and preservers (English) Author: Beasley, LeRoy B. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 69 Issue: 4 Year: 2019 Pages: 1123-1131 Summary lang: English . Category: math . Summary: Let $m$ and $n$ be positive integers, and let $R = (r_1, \ldots , r_m)$ and $S = (s_1,\ldots , s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-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 $A\in A(R,S)$. The discrepancy of A, ${\rm disc}(A)$, 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 $m\times n$ 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. (English) Keyword: Ferrers matrix Keyword: row-dense matrix Keyword: discrepancy Keyword: linear preserver Keyword: strong linear preserver MSC: 05B20 MSC: 05C50 MSC: 15A04 MSC: 15A21 MSC: 15A86 idZBL: 07144881 idMR: MR4039626 DOI: 10.21136/CMJ.2019.0092-18 . Date available: 2019-11-28T08:53:38Z Last updated: 2022-01-03 Stable URL: http://hdl.handle.net/10338.dmlcz/147920 . Reference: [1] Beasley, L. B., Pullman, N. J.: Linear operators preserving properties of graphs.Proc. 20th Southeast Conf. on Combinatorics, Graph Theory, and Computing Congressus Numerantium 70, Utilitas Mathematica Publishing, Winnipeg (1990), 105-112. Zbl 0696.05049, MR 1041590 Reference: [2] Berger, A.: The isomorphic version of Brualdies nestedness is in P, 2017, 7 pages.Available at https://arxiv.org/abs/1602.02536v2. Reference: [3] Berger, A., Schreck, B.: The isomorphic version of Brualdi's and Sanderson's nestedness.Algorithms (Basel) 10 (2017), Paper No. 74, 12 pages. Zbl 06916733, MR 3708470, 10.3390/a10030074 Reference: [4] Brualdi, R. A., Sanderson, G. J.: Nested species subsets, gaps, and discrepancy.Oecologia 119 (1999), 256-264. 10.1007/s004420050784 Reference: [5] Brualdi, R. A., Shen, J.: Discrepancy of matrices of zeros and ones.Electron. J. Comb. 6 (1999), Research Paper 15, 12 pages. Zbl 0918.05029, MR 1674136 Reference: [6] Motlaghian, S. M., Armandnejad, A., Hall, F. J.: Linear preservers of row-dense matrices.Czech. Math. J. 66 (2016), 847-858. Zbl 06644037, MR 3556871, 10.1007/s10587-016-0296-4 .

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