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 $Π$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus Π$ 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 $Π$ 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 $NO^+(8,2)$.

💡 Research Summary

The paper investigates a family of strongly regular graphs (SRGs) that arise from the geometry of hyperbolic quadrics in finite projective spaces. Let (Q^{+}(2n+1,q)) denote a non‑degenerate hyperbolic quadric in (\mathrm{PG}(2n+1,q)) and fix a generator (\Pi) (an (n)-dimensional maximal totally isotropic subspace). The vertex set of the graph (\mathcal{G}_n) consists of all points of the quadric that are not contained in (\Pi). Two distinct vertices (P,Q) are declared adjacent if either the line (\langle P,Q\rangle) is a secant of the quadric (i.e., meets the quadric in exactly two points) or the line lies entirely in the quadric and meets the fixed generator (\Pi) in at least one point. The adjacency relation is the union of these two conditions.

The authors first compute the basic parameters of (\mathcal{G}_n). The number of vertices is \