Random path in a Supersingular Isogeny Graph are used to find cycles. I want to know which properties are relevant to know if that path belong to a cycle. For example, the lenght of the path (maybe log p), the types of the nodes, the types of the edges, the amount of nodes that contains self cycles, or the amount of nodes that are related given a particular formula, etc.
I think the question needs a bit more constraints to it - every path belongs to some cycle, but maybe it's more interesting to detect when a path belongs to a cycle of fixed length or something like that? That's a question where the answer could be yes or no - small cycles are relatively rare. For precise cycle counting, maybe see this paper: Preprint Orientations and cycles in supersingular isogeny graphs
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Sarah Arpin very thanks for your answer. I returned to your paper one more time and realized that you define the number of cycles that the path belongs to (Theorem 2, equation 1). Is there a way that I can compute that number given a random path in a small graph, for example p = 100000081?
About your first question, it is really interesting. What about the paths that have the lower number of cycles that they belong to ? What is the size of the related cycles ?
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