New Strongly Regular Graphs from Finite Geometries via Switching

We show that the strongly regular graph on non-isotropic points of one type of the polar spaces of type $U(n,2)$, $O(n,3)$, $O(n,5)$, $O^{+}(n,3)$, and $O^{-}(n,3)$ are not determined by its parameters for $n\ge 6$. We prove this by using a variation of Godsil-McKay switching recently described by Wang, Qiu, and Hu. This also results in a new, shorter proof of a previous result of the first author which showed that the collinearity graph of a polar space is not determined by its spectrum. The same switching gives a linear algebra explanation for the construction of a large number of non-isomorphic designs.