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 {\displaystyle v} and an edge label i {\displaystyle i} , the rotation map returns the i {\displaystyle i} 'th neighbor of v {\displaystyle v} and the edge label that would lead back to v {\displaystyle v} .

For a D-regular graph G, the rotation map R o t G : [ N ] × [ D ] → [ N ] × [ D ] {\displaystyle \mathrm {Rot} _{G}:[N]\times [D]\rightarrow [N]\times [D]} is defined as follows: R o t G ( v , i ) = ( w , j ) {\displaystyle \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.

From the definition we see that R o t G {\displaystyle \mathrm {Rot} _{G}} is a permutation, and moreover R o t G ∘ R o t G {\displaystyle \mathrm {Rot} _{G}\circ \mathrm {Rot} _{G}} is the identity map ( R o t G {\displaystyle \mathrm {Rot} _{G}} is an involution).

• 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 π {\displaystyle \pi } -consistent if ∀ v   R o t G ( v , i ) = ( v [ i ] , π ( i ) ) {\displaystyle \forall v\ \mathrm {Rot} _{G}(v,i)=(v[i],\pi (i))} . From the definition, a π {\displaystyle \pi } -consistent rotation map is consistently labeled.

• Zig-zag product
• Rotation system