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A rotor-router walk is a deterministic version of a random walk, in which thewalker is routed to each of the neighbouring vertices in some fixed cyclicorder. We consider here directed covers of graphs (called also periodic trees)and we study several quantities related to rotor-router walks on directedcovers. The quantities under consideration are: order of the rotor-routergroup, order of the root element in the rotor-router group and the connectionwith random walks.

The rotor-router group of directed covers of graphs