On the number of cycles in a graph with restricted cycle lengths

Let L be a set of positive integers. We call a (directed) graph G an L-cycle graph if all cycle lengths in G belong to L. Let c(L, n) be the maximum number of cycles possible in an n-vertex L-cycle graph (we use c(L, n) for the number of cycles in directed graphs). In the undirected case we show that for any fixed set L, we have c(L, n) = Θ(nk/), where k is the largest element of L and 2 is the smallest even element of L (if L contains only odd elements, then c(L, n) = Θ(n) holds). We also give a characterization of L-cycle graphs when L is a single element. In the directed case we prove that for any fixed set L, we have c(L, n) = (1 + o(1))(ⁿ_k−⁻₁¹ )^k−¹, where k is the largest element of L. We determine the exact value of c((k), n) for every k and characterize all graphs attaining this maximum.