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

TL;DR

This paper characterizes divisible design graphs with specific parameters related to the 4×n-lattice graph, proving uniqueness for odd n and providing a complete classification for even n, including related parameter sets.

Contribution

It proves that for odd n, the only DDGs with these parameters are the 4×n-lattice graphs, and fully characterizes all such DDGs for even n, expanding understanding of their structure.

Findings

For odd n, the DDGs are only the 4×n-lattice graphs.

For even n, all DDGs with these parameters are classified.

Characterization of DDGs with related parameters linked to 4×n-lattice graphs.

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.