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

Definition

For a D-regular graph G, the rotation map ${R o t}_{G} : [N] \times [D] \to [N] \times [D]$ is defined as follows: ${R o t}_{G} (v, i) = (w, j)$ 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 ${R o t}_{G}$ is a permutation, and moreover ${R o t}_{G} \circ {R o t}_{G}$ is the identity map ( ${R o t}_{G}$ 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 $π$ -consistent if $\forall v {R o t}_{G} (v, i) = (v [i], π (i))$ . From the definition, a $π$ -consistent rotation map is consistently labeled.

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

Rotation map

Definition

Basic properties

Special cases and properties

See also

References