Strongly regular graphs from hyperbolic quadrics and their maximal cliques

Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $\Pi$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus \Pi$ and two vertices adjacent if they either span a secant to $Q^+(2n+1,q)$ or a line contained in $Q^+(2n+1,q)$ meeting $\Pi$ non-trivially. Then such a construction defines a strongly regular graph, which is the complement of a (non-induced) subgraph of the collinearity graph of $Q^+(2n+1,q)$. In this paper, we directly compute the parameters of $\cG_n$, which is cospectral, when $q=2$, to the tangent graph $NO^+(2n+2,2)$, but it is non-isomorphic for $n\geq3$. We also classify the maximal cliques of $\cG_3$ for $q=2$, proving as a by-product the non-isomorphism with the graph $NO^+(8,2)$.