Description
We investigate the two-graphs associated with strongly regular polar graphs that belong to three infinite classes. Specifically, we examine the strongly regular graph $\Gamma(O^\pm(2m,2))$, which has vertices representing points of a nondegenerate hyperbolic or elliptic quadric $Q^\pm(2m-1,2)$ in the projective space $PG(2m-1,2)$.
The set of vertices for $NO^\pm(2m,2)$ is the complement of $Q^\pm(2m-1,2)$. Additionally, we consider the vertices of the third graph $NO^\pm(2m+1,q)$, where $q$ is even, which correspond to hyperplanes of $PG(2m,q)$ that intersect the nondegenerate parabolic quadric in a nondegenerate hyperbolic or elliptic quadric.
Our main result is the proof of switching equivalence for the strongly regular polar graphs $NO^\pm(4m,2)$, $NO^\mp(2m+1,4)$, and $\Gamma(O^\mp(4m,2))$ with an isolated vertex.
We establish this by providing an analytic description for these graphs and their corresponding two-graphs.