2 problems

Kirkland–Šmigoc's conjecture. Then $A$ is permutation similar to a matrix whose directed graph $\Gamma$ satisfies the conditions of Proposition 7.1 in Kirkland and Šmigoc. Kirkland…

Let $DS_n$ be the set of complex numbers that occur as eigenvalues of $n$-by-$n$ doubly stochastic matrices. Let $PM_n$ be the union of the convex hulls of the $k$-th roots of unit…