Potentially eventually positive broom sign patterns

다국어 초록 (Multilingual Abstract)

A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to allow an eventually positive matrix or be potentially eventually positive if there exist at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. 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 are two open problems. In this article, we focus on the potential eventual positivity of broom sign patterns. We identify all the minimal potentially eventually positive broom sign patterns. Consequently, we classify all the potentially eventually positive broom sign patterns.

참고문헌 (Reference)

1 M. Catral, "Sign patterns that require or allow power-positivity" 19 : 121-128, 2009

2 E. M. Ellison, "Sign patterns that require eventual positivity or require eventual nonnegativity" 19 : 98-107, 2009

3 A. Berman, "Sign patterns that allow eventual positivity" 19 : 108-120, 2009

4 D. Noutsos, "Reachability and holdability of nonnegative states" 30 (30): 700-712, 2008

5 B.-L. Yu, "Potentially eventually positive star sign patterns" 31 : 541-548, 2016

6 B.-L. Yu, "Potentially eventually positive double star sign patterns" 25 (25): 1619-1624, 2012

7 B.-L. Yu, "Potential eventual positivity of sign patterns with the underlying broom graph" 23 (23): 231-241, 2017

8 B.-L. Yu, "On minimal potentially power-positive sign patterns" 6 (6): 159-167, 2012

9 S. Sriram, "On eccentric graphs of broom graphs" 5 : 146-152, 2014

10 R. A. Horn, "Matrix Analysis" Cambridge University Press 2013

11 R. A. Brualdi, "Matrices of Sign-Solvable Linear Systems" Cambridge University Press 1995

12 B.-L. Yu, "Eventual positivity of tridiagonal sign patterns" 62 (62): 853-859, 2014

13 M. Archer, "Constructions of potentially eventually positive sign patterns with reducible positive part" 4 (4): 405-410, 2011

14 R. A. Brualdi, "Combinatorial Matrix Theory" Cambridge University Press 1991