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 λ1 common neighbors, and two vertices from different classes have exactly λ2 common neighbors. 4 × n-lattice graph is the line graph of K4,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 × 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 × n-lattice graphs.