Edge version of the inducibility via the entropy method

The inducibility of a graph H is about the maximum number of induced copies of H in a graph on n vertices. We consider its edge version, that is, the maximum number of induced copies of H in a graph with m edges. Let $c(G,H)$ be the number of induced copies of H in G and ρ(H,m) = max (G,H) |E(G)| = m\. For any graph H, we prove that ρ(H,m) = Θ(mαf(H)) where αf(H) is the fractional independence number of H. Therefore, we now focus on the constant factor in front of mαf(H). In this paper, we give some results of $ρ(H,m)$ when H is a cycle or path. We conjecture that for any cycle Cₖ with k ≥ 5, ρ(Cₖ,m)= (1+o(1))( m/k)k/2 and the bound achieves by the blow up of Cₖ. For even cycles, we establish an upper bound with an extra constant factor. For odd cycles, we can only establish an upper bound with an extra factor depending on k. We prove that ρ(P₂ₗ,m) ≤ mˡ2(l-1)ˡ⁻¹ and ρ(P₂ₗ₊₁,m) ≤ mˡ⁺¹4lˡ, where l ≥ 2. We also conjecture the asymptotic value of ρ(Pₖ, m). The entropy method is mainly used to prove our results.