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

Abstract: 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. $4\times n$-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 $4\times n$-lattice graph. If $n$ is even we characterise all DDGs with such parameters. Moreover, we characterise all DDGs with parameters $(4n,3n-2,3n-6,2n-2,4,n)$ which are related to $4\times n$-lattice graphs.