Rotation map

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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 and an edge label , the rotation map returns the 'th neighbor of and the edge label that would lead back to .

Definition[]

For a D-regular graph G, the rotation map is defined as follows: 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 is a permutation, and moreover is the identity map ( 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 -consistent if . From the definition, a -consistent rotation map is consistently labeled.

See also[]

References[]

  • Reingold, O.; Vadhan, S.; Widgerson, A. (2000), "Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors", 41st Annual Symposium on Foundations of Computer Science: 3–13, arXiv:math/0406038, doi:10.1109/SFCS.2000.892006, ISBN 978-0-7695-0850-4
  • 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: 457, doi:10.1145/1132516.1132583, ISBN 978-1595931344
  • 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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