For a subgroup G of the automorphism group acting on a graph, we say that a cycle is G-consistent if there exists an element of G that acts on the cycle as a 1-step rotation. In this thesis, we will be studying the number of orbits of consistent cycles for finite and locally finite graphs. We will define their overlap and multiplicity. The group of automorphisms of a graph can be equipped with a metric, for which it is a topological group. For its closed subgroups, we can show that the stabilizer of every vertex is compact and find a formula to calculate the number of orbits of consistent cycles. We will also define the overlap tree of a graph and develop an algorithm to find it.
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