Divisible design graphs with parameters $(4n,n+2,n-2,2,4,n)$ and $(4n,3n-2,3n-6,2n-2,4,n)$

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two vertices from different classes have exactly $\lambda_2$ common neighbors. A $4$-by-$n$-lattice graph is the line graph of $K_{4,n}$. This graph is a DDG with parameters $(4n,n+2,n-2,2,4,n)$. In the paper, we consider DDGs with these parameters. We prove that if $n$ is odd, then such graph can only be a $4$-by-$n$-lattice graph. If $n$ is even, we characterise all DDGs with such parameters. Moreover, we characterise all DDGs with parameters $(4n,3n-2,3n-6,2n-2,4,n)$ that are related to $4$-by-$n$-lattice graphs. Also, we prove that if Deza graph with parameters $(4n,n+2,n-2,2)$ or $(4n,3n-2, 3n-6, 2n-2)$ is not a DDG, then $n\leq 8$. All such Deza graphs were classified by computer search.