Given a graph $H$. For a fixed edge $e$ in a complete graph $K_n$, how many copies of $H$ containing $e$ in $K_n$? Assume $n$ is large enough?
I am wondering if $n^{v_H-2}$ is an upper bound? Since an upper bound I can come up with is $\binom{n-2}{v_H-2}v_H!$. But it seems larger than $n^{v_H-2}$ when $v_H\ge 3$ and $n$ large enough.