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

Definition

For a D-regular graph G, the rotation map [math]\displaystyle{ \mathrm{Rot}_G : [N] \times [D] \rightarrow [N] \times [D] }[/math] is defined as follows: [math]\displaystyle{ \mathrm{Rot}_G (v,i)=(w,j) }[/math] 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 [math]\displaystyle{ \mathrm{Rot}_G }[/math] is a permutation, and moreover [math]\displaystyle{ \mathrm{Rot}_G \circ \mathrm{Rot}_G }[/math] is the identity map ([math]\displaystyle{ \mathrm{Rot}_G }[/math] 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 [math]\displaystyle{ \pi }[/math]-consistent if [math]\displaystyle{ \forall v \ \mathrm{Rot}_G(v,i)=(v[i],\pi (i)) }[/math]. From the definition, a [math]\displaystyle{ \pi }[/math]-consistent rotation map is consistently labeled.

See also

• Zig-zag product
• Rotation system

References

• Reingold, O.; Vadhan, S.; Widgerson, A. (2000). "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors". Proceedings 41st Annual Symposium on Foundations of Computer Science. pp. 3–13. doi:10.1109/SFCS.2000.892006. ISBN 978-0-7695-0850-4. 

• Reingold, O (2008), "Undirected connectivity in log-space", Journal of the ACM 55 (4): 1–24, doi:10.1145/1391289.1391291 

• Reingold, O.; Trevisan, L.; Vadhan, S. (2006), "Pseudorandom walks on regular digraphs and the RL vs. L problem", Proceedings of the thirty-eighth annual ACM symposium on Theory of Computing, pp. 457–466, doi:10.1145/1132516.1132583, ISBN 978-1595931344 

• Alexander, C. (2021), A Note on Consistent Rotation Maps of Graph Cartesian Products, doi:10.13140/RG.2.2.19721.57446 

• Alexander, C. (2021), Consistent Rotation Maps Induce a Unitary Shift Operator in Discrete Time Quantum Walks, doi:10.13140/RG.2.2.17614.59201