2025/03/18 by Kaarnioja, Vesa, Zepernick, André-Alexander
#06A12 #11C20 #15B34 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.14180
Let Kn denote the set of all nonsingular n× n lower triangular (0,1)-matrices. Hong and Loewy (2004) introduced the number sequence cn=min\λ|λ~is an eigenvalue of~XX\rm T,~X∈ Kn\, n∈\mathbb Z+. There have been a number of attempts in the literature to obtain bounds on the numbers cn by Mattila (2015), Altinisik et al. (2016), Kaarnioja (2021), Loewy (2021), and Altinisik (2021). In this paper, improved upper and lower bounds are derived for the numbers cn. By considering the characteristic polynomial corresponding to the matrix Zn satisfying cn=‖Zn‖2-1, it is shown that the second largest eigenvalue of Zn is bounded from above by \frac45 leading to an improved upper bound on cn. On the other hand, Samuelson's inequality applied to the roots of the characteristic polynomial of Zn yields an improved lower bound. Numerical experiments demonstrate the quality of the new bounds.