Bounds on the spectrum of nonsingular triangular $(0,1)$-matrices

Let $K_n$ be the set of all nonsingular $n\times n$ lower triangular $(0,1)$-matrices. Hong and Loewy (2004) introduced the numbers $$c_n=\mathrm{m}\mathrm{i}\mathrm{n}\{\lambda \mid \lambda \text{ is an eigenvalue of }X{X}^{\mathrm{T}},X\in K_n\},n\in {\mathbb{Z}}_{+}\mathrm{.}$$ A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008): $$C_n=\mathrm{m}\mathrm{a}\mathrm{x}\{\lambda \mid \lambda \text{ is an eigenvalue of }X{X}^{\mathrm{T}},X\in K_n\},n\in {\mathbb{Z}}_{+}\mathrm{.}$$ These numbers can be used to bound the singular values of matrices belonging to $K_n$ and they appear, e.g., in eigenvalue bounds for power GCD matrices, lattice-theoretic meet and join matrices, and related number-theoretic matrices. In this paper, it is shown that for $n$ odd, one has the lower bound $$c_n\ge \frac{1}{\sqrt{\frac{1}{25}{\phi }^{-4n}+\frac{2}{25}{\phi }^{-2n}-\frac{2}{5\sqrt{5}}n{\phi }^{-2n}-\frac{23}{25}+n+\frac{2}{25}{\phi }^{2n}+\frac{2}{5\sqrt{5}}n{\phi }^{2n}+\frac{1}{25}{\phi }^{4n}}},$$ and for $n$ even, one has $$c_n\ge \frac{1}{\sqrt{\frac{1}{25}{\phi }^{-4n}+\frac{4}{25}{\phi }^{-2n}-\frac{2}{5\sqrt{5}}n{\phi }^{-2n}-\frac{2}{5}+n+\frac{4}{25}{\phi }^{2n}+\frac{2}{5\sqrt{5}}n{\phi }^{2n}+\frac{1}{25}{\phi }^{4n}}},$$ where $\phi$ denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altini\c{s}ik et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that $c_n\sim 5{\phi }^{-2n}$ as $n\to \mathrm{\infty }$. In addition, a new closed form expression is derived for the numbers $C_n$, viz. $$C_n=\frac{1}{4}{\csc }^2\left(\frac{\pi }{4n+2}\right)=\frac{4n^2}{{\pi }^2}+\frac{4n}{{\pi }^2}+(\frac{1}{12}+\frac{1}{{\pi }^2})+\mathcal{O}\left(\frac{1}{n^2}\right),n\in {\mathbb{Z}}_{+}\mathrm{.}$$