In a graph, I understand a cycle to be a traversal from Node A, traversing each (but not every) vertex once, and returning to Node A. Now I THINK a distinct cycle is where they don't share any vertices, but I might be wrong. Can someone clear this up for me?
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(Expanding the comment by Brian M. Scott): being distinct is not a property of a cycle, but a relation between two cycles. Two cycles are distinct if they are not the same cycle.
Usage example: "For all $n\ge 3$, the number of distinct Hamilton cycles in the complete graph $K_n$ is $(n−1)!/2$."