Estimating Global Subgraph Counts by Sampling

We give a simple proof of generalization an inequality for homomorphism counts by Sidorenko (1994). A special case our says that if $d_v$ denotes the degree vertex $v$ in graph $G$ and $\textrm{Hom}_\Delta(H,G)$ number homomorphisms from connected $H$ on $h$ vertices to which map particular with $d_v \ge \Delta$, then $\textrm{Hom}_\Delta(H,G) \le \sum_{v\in G} d_v^{h-1}\mathbf{1}_{d_v\ge \Delta}$.
 use this study minimum sample size needed estimate copies sampling at random.