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 $\rho(H,m) = \max \{c(G,H) \mid |E(G)| = m\}$. For any graph $H$, we prove that $\rho(H,m) = \Theta(m^{\alpha_f(H)})$ where $\alpha_f(H)$ is the fractional independence number of $H$. Therefore, we now focus on the constant factor in front of $m^{\alpha_f(H)}$. In this paper, we give some results of $\rho(H,m)$ when $H$ is a cycle or path. We conjecture that for any cycle $C_k$ with $k \ge 5$, $\rho(C_k,m)= (1+o(1))\left( m/k\right)^{k/2}$ and the bound is achieved by the blow up of $C_k$. 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 $\rho(P_{2l},m) \le \frac{m^l}{2(l-1)^{l-1}}$ and $\rho(P_{2l+1},m) \le \frac{m^{l+1}}{4l^l}$, where $l \ge 2$. We also conjecture the asymptotic value of $\rho(P_k, m)$. The entropy method is mainly used to prove our results.