Strongly regular graphs in hyperbolic quadrics

TL;DR

This work constructs a family of strongly regular graphs $\mathcal{G}_n$ from hyperbolic quadrics $Q^+(2n+1,q)$ by deleting a generator $\Pi$ and declaring adjacency when lines are secants or meet $\Pi$. For general $q$, $\mathcal{G}_n$ is SRG with explicit parameters, and when $q=2$ it is cospectral with the known graph $NO^{+}(2n+2,2)$ but non-isomorphic for all $n\ge 3$. The authors provide a complete geometric characterization, including a detailed clique classification for $\mathcal{G}_3$ via computational methods, which together yield a robust isomorphism distinction and deeper insight into the clique structure of these graphs. The results extend understanding of SRGs arising from finite classical polar spaces and illustrate how cospectral graphs can be distinguished by subtle geometric configurations. The work also highlights potential avenues for exploring isomorphism classes and switching behavior among cospectral SRGs tied to hyperbolic quadrics.

Abstract

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 prove the non-isomorphism by analyzing the case of the quadric $Q^+(7,2)$.

Theorems & Definitions (33)

• Theorem 3.1
• proof
• Remark 3.2
• Proposition 4.1
• proof
• Corollary 4.3
• proof
• Lemma 4.4
• proof
• Lemma 4.5