Rotation map

Not to be confused with Rotation (mathematics).

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.

## Contents

* 1 Definition
* 2 Basic properties
* 3 Special cases and properties
* 4 See also
* 5 References

## Definition

For a D-regular graph G, the rotation map RotG:[N]×[D]->[N]×[D] is defined as follows: RotG(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 RotG is a permutation, and moreover RotG∘RotG is the identity map (RotG 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 π-consistent if ∀v RotG(v,i)=(v[i],π(i)). From the definition, a π-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

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