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The region in the complex plane containing the eigenvalues of all stochasticmatrices of order n was described by Karpelevic in 1988, and it is since thenknown as the Karpelevic region. The boundary of the Karpelevic region is theunion of disjoint arcs called the Karpelevic arcs. We provide a completecharacterization of the Karpelevic arcs that are powers of some otherKarpelevic arc. Furthermore, we find the necessary and sufficient conditionsfor a sparsest stochastic matrix associated with the Karpelevic arc of order nto be a power of another stochastic matrix.

Powers of Karpelevic arcs and their Sparsest Realising matrices