Small subgraphs of the random graph

Question

If I look at the distribution of the number of small subgraphs in the random graph isomorphic to a connected graph $H$, this is asymptotically Poisson.

What proportion of these small subgraphs intersect another, fixed subgraph?

Also, what if I want them to intersect in a specific way, not just arbitrarily? Do the subgraphs occur on vertices that are selected uniformly at random, so the `striking' probability is just the probability I select those vertex, chosen uniformly at random?

Answer 1

Given $G,H$ two finite connected graphs, let $H_1,...,H_k$ be all the connected graphs obtained as a union of $G$ and $H$ (the connectedness of the $H_i$ implies that $H$ and $G$ intersect). Then making the sum for $i$ of the occurences of $H_i$ as a subgraph, for $i$ between $1$ and $k$, gives the answer to the first question. For the second question simply count the number of occurences of the $H_i$ that you like. Sorry I did not understand the last question.