Strongly regular graphs from orthogonal groups $O^+(6,2)$ and $O^-(6,2)$

TL;DR

This paper constructs all strongly regular graphs with up to 600 vertices admitting certain orthogonal group actions, discovering new graphs and establishing methods to derive new graphs from orbit matrices.

Contribution

It provides a complete classification of such graphs for specific groups and introduces a novel approach to generate new strongly regular graphs from orbit matrices.

Findings

Existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68).

Construction of a previously unknown strongly regular graph with parameters (540,224,88,96).

Demonstration that orbit matrices can be used to derive new strongly regular graphs.

Abstract

In this paper we construct all strongly regular graphs, with at most 600 vertices, admitting a transitive action of the orthogonal group $O^+(6,2)$ or $O^-(6,2)$. Consequently, we prove the existence of strongly regular graphs with parameters (216,40,4,8) and (540,187,58,68). We also construct a strongly regular graph with parameters (540,224,88,96) that was to the best of our knowledge previously unknown. Further, we show that under certain conditions an orbit matrix $M$ of a strongly regular graph $\Gamma$ can be used to define a new strongly regular graph $\widetilde{\Gamma}$, where the vertices of the graph $\widetilde{\Gamma}$ correspond to the orbits of $\Gamma$ (the rows of $M$). We show that some of the obtained graphs are related to each other in a way that one can be constructed from an orbit matrix of the other.