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

Definition

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

Basic properties

From the definition we see that ${\displaystyle \mathrm {Rot} _{G}}$ is a permutation, and moreover ${\displaystyle \mathrm {Rot} _{G}\circ \mathrm {Rot} _{G}}$ is the identity map (${\displaystyle \mathrm {Rot} _{G}}$ 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 ${\displaystyle \pi }$-consistent if ${\displaystyle \forall v\ \mathrm {Rot} _{G}(v,i)=(v,\pi (i))}$. From the definition, a ${\displaystyle \pi }$-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. arXiv:math/0406038. doi:10.1109/SFCS.2000.892006. ISBN 978-0-7695-0850-4. S2CID 420651.

• Reingold, O (2008), "Undirected connectivity in log-space", Journal of the ACM, 55 (4): 1–24, doi:10.1145/1391289.1391291, S2CID 207168478

• 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, S2CID 17360260

• 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

• Extensions and generalizations of graphs
• Graph operations