# Isogeny and Ramanujan Graphs

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**I (Jack) know nothing about this. At all. But it will need to be talked about.**
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* Isogeny graph (general definition, degree, duals...).
* Starting vertex (Bröker's algorithm).
* Isogeny Volcanos: Sutherland might be a good source <https://arxiv.org/abs/1208.5370>.
* Supersingular isogeny graphs
  * Size, everything is defined over GF(p^2). (\*as long as the degree divides (p+1)^2 or (p-1)^2).
  * Random walks are probably the best motivation to define Ramanujanness, and are directly applicable to cryptography. A (perhaps too large) source is Hoory-Linial-Wigderson.
  * Consequence from Ramanujan + random walk convergence: O(log p) diameter.