Rotation map

In mathematics, a rotation map is a function that represents an undirected edge-labeled graph, where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson (“Entropy waves, the zig-zag graph product, and new constant-degree expanders”, 2002) in order to conveniently define the zig-zag product and prove its properties. Given a vertex $$v$$ and an edge label $$i$$, the rotation map returns the $$i$$'th neighbor of $$v$$ and the edge label that would lead back to $$v$$.

Definition
For a D-regular graph G, the rotation map $$\mathrm{Rot}_G : [N] \times [D] \rightarrow [N] \times [D]$$ is defined as follows: $$\mathrm{Rot}_G (v,i)=(w,j)$$ if the i th edge leaving v leads to w, and the j th edge leaving w leads to v.

Basic properties
From the definition we see that $$\mathrm{Rot}_G$$ is a permutation, and moreover $$\mathrm{Rot}_G \circ \mathrm{Rot}_G$$ is the identity map ($$\mathrm{Rot}_G$$ is an involution).

Special cases and properties

 * A rotation map is consistently labeled if all the edges leaving each vertex are labeled in such a way that at each vertex, the labels of the incoming edges are all distinct. Every regular graph has some consistent labeling.
 * A consistent rotation map can be used to encode a coined discrete time quantum walk on a (regular) graph.
 * A rotation map is $$\pi$$-consistent if $$\forall v \ \mathrm{Rot}_G(v,i)=(v[i],\pi (i))$$. From the definition, a $$\pi$$-consistent rotation map is consistently labeled.