Switching for Small Strongly Regular Graphs

TL;DR

This paper introduces new strongly regular graphs with specific parameters using switching techniques, significantly expanding the known examples and providing insights into their automorphism groups.

Contribution

It applies Godsil-McKay and Wang-Qiu-Hu switching methods to generate numerous new SRGs, including a large set with parameters (81,30,9,12), and offers statistical analysis of automorphism groups.

Findings

Found over 16 million SRGs with parameters (81,30,9,12)

Discovered the third construction method for Krčadinac partial geometry

Provided automorphism group size statistics for the generated graphs

Abstract

We provide an abundance of strongly regular graphs (SRGs) for certain parameters $(n, k, \lambda, \mu)$ with $n < 100$. For this we use Godsil-McKay (GM) switching with a partition of type $4,n-4$ and Wang-Qiu-Hu (WQH) switching with a partition of type $3,3,n-6$ or $4,4,n-8$. In most cases, we start with a highly symmetric graph which belongs to a finite geometry. Many of the obtained graphs are new; for instance, we find 16565438 strongly regular graphs with parameters $(81, 30, 9, 12)$ while only 15 seem to be described in the literature. We provide statistics about the size of the occurring automorphism groups. We also find the recently discovered Kr\v{c}adinac partial geometry, thus finding a third method of constructing it.