Browse > Article
http://dx.doi.org/10.4134/BKMS.b180210

ISOLATION NUMBERS OF INTEGER MATRICES AND THEIR PRESERVERS  

Beasley, LeRoy B. (Department of Mathematics and Statistics Utah State University)
Kang, Kyung-Tae (Department of Mathematics Jeju National University)
Song, Seok-Zun (Department of Mathematics Jeju National University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 535-545 More about this Journal
Abstract
Let A be an m × n matrix over nonnegative integers. The isolation number of A is the maximum number of isolated entries in A. We investigate linear operators that preserve the isolation number of matrices over nonnegative integers. We obtain that T is a linear operator that strongly preserve isolation number k for 1 ≤ k ≤ min{m, n} if and only if T is a (P, Q)-operator, that is, for fixed permutation matrices P and Q, T(A) = P AQ or, m = n and T(A) = P AtQ for any m × n matrix A, where At is the transpose of A.
Keywords
Isolation number; upper ideal; linear operator; (P, Q)-operator;
Citations & Related Records
연도 인용수 순위
  • Reference
1 L. B. Beasley, Isolation number versus Boolean rank, Linear Algebra Appl. 436 (2012), no. 9, 3469-3474. https://doi.org/10.1016/j.laa.2011.12.013   DOI
2 L. B. Beasley, Preservers of upper ideals of matrices: Tournaments; Primitivity, J. Combin. Math. Combin. Comput. 100 (2017), 55-75.
3 L. B. Beasley, D. A. Gregory, and N. J. Pullman, Nonnegative rank-preserving operators, Linear Algebra Appl. 65 (1985), 207-223. https://doi.org/10.1016/0024-3795(85)90098-9   DOI
4 L. B. Beasley, Y. B. Jun, and S.-Z. Song, Possible isolation number of a matrix over nonnegative integers, Czechoslovak Math. J. 68(143) (2018), no. 4, 1055-1066. https://doi.org/10.21136/CMJ.2018.0068-17
5 L. B. Beasley and N. J. Pullman, Boolean-rank-preserving operators and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984), 55-77. https://doi.org/10.1016/0024-3795(84)90158-7   DOI
6 L. B. Beasley and N. J. Pullman, Linear operators that strongly preserve upper ideals of matrices, Congr. Numer. 88 (1992), 229-243.
7 J. A. Bondy and U. S. R. Murty, Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008. https://doi.org/10.1007/978-1-84628-970-5
8 D. Gregory, N. J. Pullman, K. F. Jones, and J. R. Lundgren, Biclique coverings of regular bigraphs and minimum semiring ranks of regular matrices, J. Combin. Theory Ser. B 51 (1991), no. 1, 73-89. https://doi.org/10.1016/0095-8956(91)90006-6   DOI
9 R. A. Brualdi and H. J. Ryser, Combinatorial matrix theory, Encyclopedia of Mathematics and its Applications, 39, Cambridge University Press, Cambridge, 1991. https://doi.org/10.1017/CBO9781107325708
10 D. de Caen, D. A. Gregory, and N. J. Pullman, The Boolean rank of zero-one matrices, in Proceedings of the Third Caribbean Conference on Combinatorics and Computing (Bridgetown, 1981), 169-173, Univ. West Indies, Cave Hill Campus, Barbados, 1981.
11 K.-T. Kang, S.-Z. Song, and L. B. Beasley, Linear preservers of term ranks of matrices over semirings, Linear Algebra Appl. 436 (2012), no. 7, 1850-1862. https://doi.org/10.1016/j.laa.2011.08.046   DOI
12 K. H. Kim, Boolean Matrix Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics, 70, Marcel Dekker, Inc., New York, 1982.
13 S. Pierce et al., A survey of linear preserver problems, Linear and Multilinear Algebra 33 (1992), 1-119.   DOI
14 G. Markowsky, Ordering D-classes and computing Schein rank is hard, Semigroup Forum 44 (1992), no. 3, 373-375. https://doi.org/10.1007/BF02574357   DOI