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

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 $\tilde{\Gamma }$, where the vertices of the graph $\tilde{\Gamma }$ 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.

14 pages, some remarks added to the first version