AUTHORS: Ber-Lin Yu, Hong-Mei Bao, Jie Cui
Download as PDF
ABSTRACT: A sign pattern is a matrix whose entries belong to the set {+,−,0}. An n-by-n sign pattern A is said to allow an eventually positive matrix or be potentially eventually positive if there exist some real matrices A with the same sign pattern as A and a positive integer k0 such that Ak > 0 for all k ≥ k0. Identifying the necessary and sufficient conditions for an n-by-n sign pattern to be potentially eventually positive, and classifying the n-by-n sign patterns that allow an eventually positive matrix were posed as two open problems by Berman, Catral, Dealba, et al. In this article, a new tree sign pattern A obtained from one tridiagonal sign pattern by adding one pendent edge are investigated. Some necessary conditions for the sign pattern A to allow an eventually positive matrix are established first. Then all the minimal tree sign patterns that allow an eventually positive matrix are identified. Finally, all the tree sign patterns that allow an eventually positive matrix are classified.
KEYWORDS: Sign pattern, Potential eventual positivity, Checkerboard block sign pattern
REFERENCES:
[1] F. Hall, Z. Li, Sign pattern matrices, in: L. Hogben(Ed.), Handbook of Linear Algebra, Boca Ration: Chapman & Hall/CRC Press, 2007.
[2] R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge: Cambridge University Press, 1991.
[3] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge: Cambridge University Press, 1995.
[4] A. Berman, M. Catral, L. M. Dealba, A. Elhashash, F. Hall, L. Hogben, I. J. Kim, D. D. Olesky, P. Tarazaga, M. J. Tsatsomeros, P. van den Driessche, Sign patterns that allow eventual positivity, Electronic Journal of Linear Algebra 19, 2010, pp. 108–120.
[5] B. -L. Yu, T. -Z. Huang, J. Luo, H. B. Hua, Potentially eventually positive double star sign patterns, Applied Mathematics Letters 25, 2012, pp. 1619–1624.
[6] B. -L. Yu, T. -Z. Huang, On minimal potentially power-positive sign patterns, Operators and Matrices 6, 2012, pp. 159–167.
[7] M. Catral, L. Hogben, D. D. Olesky, P. van den Driessche, Sign patterns that require or allow power-positivity, Electronic Journal of Linear Algebra 19, 2010, pp. 121–128.
[8] B. -L. Yu, T. -Z. Huang, C. Hong, D. D. Wang, Eventual positivity of tridiagonal sign patterns, Linear and Multilinear Algebra, 62, 2014, pp. 853–859.
[9] B. -L. Yu, J. Cui, Potential eventual positivity of one specific tree sign pattern, WSEAS Transactions on Mathematics, 15,2016, pp. 261–270.
[10] M. Archer, M. Catral, C. Erickson, R. Haber, L. Hogben, X. Martinez-Rivera, A. Ochoa, Constructions of potentially eventually positive sign patterns with reducible positive part, Involve 4, 2011, pp. 405–410.
[11] Y. Hu, X. Zhou, On the Harmonic index of the unicyclic and bicyclic graphs , WSEAS Transactions on Mathematics 12, 2013, pp. 716–726.
[12] T. Liu, Y. Hu, The 3-Rainbow index of graph operations, WSEAS Transactions on Mathematics 13, 2014, pp. 161–170.
[13] L. Zuo, F. Wu, C. Hong, S. Zhang, Equitable colorings of Cartesian product graphs of wheels with complete bipartite graphs, WSEAS Transactions on Mathematics 13, 2014, pp. 236–245.
[14] E. M. Ellison, L. Hogben, M. J. Tsatsomeros, Sign patterns that require eventual positivity or require eventual nonnegativity, Electronic Journal of Linear Algebra 19, 2010, pp. 98–107.
