I am considering the random graphs generated by the Erdős-Rényi model for this question. Random Graphs as Models of Networks by Newman is a reference on this topic. A random graph $\Gamma_{n,p}$ has $n$ vertices and there is $p$ probability that there is an edge between any pair of vertices. It is already known that the average degree of a vertex of a random graph, $z$, is $(n-1) p$. The probability distribution of the degree $k$ of a vertex is,
$$ p_k = {n \choose k} p^k (1-p)^{n-1} \simeq \frac{z^k e^{-z}}{k!} $$ which is the Poisson distribution.
My questions:
For two random graphs, $\Gamma_{n_1,p_1}$ and $\Gamma_{n_2,p_2}$, let's take a vertex pair, $(v_{1_i}, v_{2_j})$, where, $v_{1_i} \in V(\Gamma_{n_1,p_1})$, $v_{2_j} \in V(\Gamma_{n_2,p_2})$, $i = 1, 2, \ldots, |V(\Gamma_{n_1,p_1})|$, $j = 1, 2, \ldots, |V(\Gamma_{n_2,p_2})|$ and $deg(v_{1_i}) = deg(v_{2_j})$. How many such pairs are possible?
What is the probability distribution of the number of such pairs if $deg(v_{1_i}) = deg(v_{2_j}) = k$?
My effort so far:
As $\Gamma_{n_1,p_1}$ and $\Gamma_{n_2,p_2}$ are independently generated, the Poisson distributions will be mutually independent. In that case, the probability distribution of the number of such pairs if $deg(v_{1_i}) = deg(v_{2_j}) = k$, $p_{1,2,k}$, is
$$p_{1,2,k} = {n_1 \choose k} p^k_1 (1-p_1)^{n_1-1} \times {n_2 \choose k} p^k_2 (1-p_2)^{n_2-1}$$
Am I getting it right?