Consider a bipartite graph of order $2n$ with equal bipartitions $C_1$ and $C_2$, where, $$C_i = \{v_{i,1}, v_{i,2}, v_{i,3} \dots v_{i,n}\}; i = 1, 2.$$ Given two vertices $v_{i,p}$ and $v_{i,q}$, $Nbd(v_{i,p}, v_{i,q})$ denotes set vertices adjacent to both $v_{i,p}$ and $v_{i,q}$ with $1 \le p,q \le n$. Indexes $p$ and $q$ may or may not be equal. We denote cardinallity of $Nbd(v_{i,p}, v_{i,q})$ by $\#(Nbd(v_{i,p}, v_{i,q}))$.

I want to calculate number of bipartite graphs, such that, $$\#(Nbd(v_{1,p}, v_{1,q})) = \#(Nbd(v_{2,p}, v_{2,q})) ~\forall~ p,q ~\text{with}~ 1 \le p,q \le n.$$

To construct all these graphs, I am also looking for an existing algorithm, if available.