Classification of divisible design graphs with at most 39 vertices

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. A DDG with $m=1$, $n=1$, or ${\lambda }_1={\lambda }_2$ is called improper, otherwise it is called proper. We present new constructions of DDGs and, using a computer enumeration algorithm, we find all proper connected DDGs with at most 39 vertices, except for three tuples of parameters: $(32,15,6,7,4,8)$, $(32,17,8,9,4,8)$, $(36,24,15,16,4,9)$.