The rotor-router group of directed covers of graphs

TL;DR

This paper investigates the properties of rotor-router walks on directed covers of graphs, focusing on group order, root element order, and their relation to random walks, providing insights into deterministic analogs of stochastic processes.

Contribution

It introduces new analyses of rotor-router groups on directed covers of graphs, linking deterministic walks to probabilistic behavior and exploring algebraic properties.

Findings

Determined the order of the rotor-router group for directed covers.

Established the relationship between rotor-router walks and random walks on these structures.

Analyzed the algebraic properties of the rotor-router group elements.

Abstract

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