Cycles of given lengths in unicyclic components in sparse random graphs

Let L be subset of {3,4,…} and let Xn,M(L) the number cycles belonging to unicyclic components whose length is in random graph G(n,M). We find limiting distribution subcritical regime M=cn with c<1/2 critical M=n2(1+μn−1/3) μ=O(1). Depending on a condition involving series ∑ℓ∈Lzℓ/(2ℓ), we obtain limit either Poisson or normal as n→∞.