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A $k$-regular graph is called a divisible design graph (DDG for short) if itsvertex set can be partitioned into $m$ classes of size $n$, such that twodistinct vertices from the same class have exactly $\lambda_1$ commonneighbors, 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 weconsider DDGs with these parameters. We prove that if $n$ is odd then suchgraph can only be a $4\times n$-lattice graph. If $n$ is even we characteriseall DDGs with such parameters. Moreover, we characterise all DDGs withparameters $(4n,3n-2,3n-6,2n-2,4,n)$ which are related to $4\times n$-latticegraphs.

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