Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.

I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$. If I am not missing something the probability that $G'$ has $2 \leq k \leq n$ vertices is

$P[|V(G')| = k] = {e \choose n-k} (1-p)^{e-n+k}p^{n-k}$.

It then holds that $P[|V(G')| = 1] = 1 - \displaystyle \sum_{k=2}^n P[|V(G')| = k]$.

And thus $E[|V(G')|] = 1 + \displaystyle \sum_{k=2}^{n} (k-1) {e \choose n-k} p^{n-k} (1-p)^{e-n+k}$

The above sum seems quite complicated to evaulate/bound. I am sure I am not the first one that studied such a probabilistic space and since I couldn't find any estimates for $E[|V(G')|]$ in my textbook I am asking: is there any simple identity/estimate for $E[|V(G')|]$ ?

Let $G'$ be a graph obtained from $G$ after contracting each edge with probability $p$. Let $n = |V(G)|, e = |E(G)|$.

I would like to compute (or at least obtain a lower bound) for $E[|V(G')|]$ in terms of some known graph invariants (number of edges, degree sequence, connectivity,..)

I am sure I am not the first one that studied such a probabilistic space and since I couldn't find any estimates for $E[|V(G')|]$ in my textbook I am asking: is there any simple identity/estimate for $E[|V(G')|]$ ? Is there any reference to a paper studying this quantity?

Edit: I have removed the completely wrong attempt to estimate $E[|V(G')|]$.