Suppose I have a random 3 regular graph - there are many results about the expected number of cycles of given length in such a graph, and also about things like the probability that any two cycles of given length will intersect (the best resource I know of is a paper of McKay, Wormold and Wysocka, plus of course Bollobas). My question is the following - if I randomly choose an edge $e$, what is the expected number of cycles of some given length containing that edge? I guess the best way to formulate this might be to consider the set consisting of the number of (simple) cycles each edge is on, and ask is it known what the distribution of those numbers is?

Thanks