  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.
