Geodesic
Divisible design graphs with parameters $(4n,n+2,n-2,2,4,n)$ and $(4n,3n-2,3n-6,2n-2,4,n)$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1742-1756

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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.
Keywords: divisible desing graph, divisible design, Deza graph, lattice graph. 
@article{SEMR_2021_18_2_a44,
     author = {L. Shalaginov},
     title = {Divisible design graphs with parameters $(4n,n+2,n-2,2,4,n)$ and $(4n,3n-2,3n-6,2n-2,4,n)$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1742--1756},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a44/}
}
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L. Shalaginov. Divisible design graphs with parameters $(4n,n+2,n-2,2,4,n)$ and $(4n,3n-2,3n-6,2n-2,4,n)$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1742-1756. http://geodesic.mathdoc.fr/item/SEMR_2021_18_2_a44/