Strongly regular graphs from hyperbolic quadrics and their maximal cliques

Let $Q^{+}(2n+1,q)$ be a hyperbolic quadric of $\text{PG}(2n+1,q)$. Fix a generator $\mathrm{\Pi }$ of the quadric. Define ${\text{cG}}_n$ as the graph with vertex set the points of $Q^{+}(2n+1,q)\setminus \mathrm{\Pi }$ 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 $\mathrm{\Pi }$ 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 ${\text{cG}}_n$, which is cospectral, when $q=2$, to the tangent graph $N{O}^{+}(2n+2,2)$, but it is non-isomorphic for $n\ge 3$. We also classify the maximal cliques of ${\text{cG}}_3$ for $q=2$, proving as a by-product the non-isomorphism with the graph $N{O}^{+}(8,2)$.