2 problems

Let $\mathscr{S}(n,e)$ be the set of $n\times n$ $(0,1)$-matrices with exactly $e=c^2+t$ ones, where $2\leq t\leq 2c$. If $A\in\mathscr{S}(n,e)$ attains the maximum spectral radius…

Let $\widehat{\sigma}_t$ denote the spectrum under consideration, and form an augmented spectrum by adding any number of zeros to it. Bound conjecture. The value $t=\frac{1}{3}$ is…