List of constructions of $NO^+(6,2)$
Valentino Smaldore
TL;DR
The note systematically collects seven equivalent geometric and algebraic constructions of the tangent-polar graph $NO^+(6,2)$, a strongly regular graph with parameters $srg(28,15,6,10)$. It spans models based on a hyperbolic quadric $Q^+(5,2)$, the Klein correspondence, antiflags in $PG(2,2)$ with Veronese embeddings, Veronese and secant varieties, and symmetric $3\times3$ matrices, all yielding graphs isomorphic to $NO^+(6,2)$ or its complement. Each construction provides a distinct ambient space and vertex-set: $PG(5,2)\setminus Q^+(5,2)$, lines of $PG(3,2)$ under Klein, $PG(2,2)$ antiflags or conics via Veronese, and nonsingular symmetric matrices over $\mathbb{F}_q$. The work thereby connects finite geometry, association schemes, and SRG theory, enabling multiple lenses for analysis and potential applications in combinatorial design and symmetry studies.
Abstract
In this note, we list many equivalent constructions of the tangent polar graph $NO^+(6,2)$.