On the trace-zero doubly stochastic matrices of order 5

We propose a graph theoretic approach to determine trace of product of two permutation matrices through a weighted digraph representation of the permutation matrices. Consequently, we derive trace-zero doubly stochastic (DS) matrices of order $5$ whose $k$-th power is also a trace-zero DS matrix for $k\in \{2,3,4,5\}$. Then, we determine necessary conditions for the coefficients of a generic polynomial of degree $5$ to be realizable as the characteristic polynomial of a trace-zero DS matrix of order $5$. Finally, we approximate the eigenvalue region of trace-zero DS matrices of order $5.$