Classification of divisible design graphs with at most 39 vertices

TL;DR

This paper classifies and constructs divisible design graphs with up to 39 vertices, providing new examples and complete enumeration results, except for three unresolved parameter sets.

Contribution

It introduces new constructions of divisible design graphs and enumerates all proper connected cases with up to 39 vertices, excluding three specific parameter configurations.

Findings

All proper connected DDGs with ≤39 vertices found except three parameter sets.

New constructions of DDGs provided.

Enumeration results contribute to classification of DDGs.

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: $(32,15,6,7,4,8)$, $(32,17,8,9,4,8)$, $(36,24,15,16,4,9)$.