= Cycle graph
{disambiguate=algebra}
{wiki}
How to build it: https://math.stackexchange.com/questions/3137319/how-in-general-does-one-construct-a-cycle-graph-for-a-group/3162746\#3162746 good answer with <ASCII art>. You basically just pick each element, and repeatedly apply it, and remove any path that has a longer version.
Immediately gives the <generating set of a group> by looking at elements adjacent to the origin, and more generally the <order of an element of a group>[order of each element].
TODO <uniqueness>: can two different <groups> have the same cycle graph? It does not seem to tell us how every element interact with every other element, only with itself. This is in contrast with the <Cayley graph>, which more accurately describes group structure (but does not give the order of elements as directly), so feels like it won't be unique.