Powers of Karpelevic arcs and their Sparsest Realising matrices

The region in the complex plane containing the eigenvalues of all stochastic matrices of order n was described by Karpelevic in 1988, and it is since then known as the Karpelevic region. The boundary of the Karpelevic region is the union of disjoint arcs called the Karpelevic arcs. We provide a complete characterization of the Karpelevic arcs that are powers of some other Karpelevic arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevic arc of order n to be a power of another stochastic matrix.