Classification of divisible design graphs with at most 39 vertices

A k-regular graph is called a divisible design graph (DDG) 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 neighbours, and two vertices from different classes have exactly lambda 2 common neighbours. 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 ), and ( 36 , 24 , 15 , 16 , 4 , 9 ).