On a graph isomorphic to $NO^{+}(6,2)$

TL;DR

This paper constructs a new graph based on the secant variety of the Veronese surface in PG(5,2) and proves it is isomorphic to the tangent graph of a hyperbolic quadric, revealing a novel geometric graph isomorphism.

Contribution

It introduces a graph derived from the secant variety of the Veronese surface and establishes its isomorphism with the tangent graph of a hyperbolic quadric in PG(5,2).

Findings

The graph has 28 vertices.

The graph is isomorphic to NO^{+}(6,2).

The construction links algebraic geometry with graph theory.

Abstract

Let $Q^{+}(2n-1,2)$ be a non-degenerate hyperbolic quadric of $PG(2n-1,2)$. Let $NO^{+}(2n,2)$ be the tangent graph, whose vertices are the points of $PG(2n-1,2) \setminus Q^{+}(2n-1,2)$ and two vertices $u,~v$ are adjacent if the line joining $u$ and $v$ is tangent to $Q^{+}(2n-1,2)$. Then $NO^{+}(2n-1,q)$ is a strongly regular graph. Let $\mathcal{V}^{4}_{2}$ be the \textit{Veronese surface} in $PG(5,q)$, and $\mathcal{M}^{3}_{4}$ its \textit{secant variety}. When $q=2$, $|Q^{+}(5,2)|=|\mathcal{M}^{3}_{4}|=35$. In this paper we define the graph $N\mathcal{M}^{3}_{4}$, with 28 vertices in $PG(5,2)\setminus\mathcal{M}^{3}_{4}$ and with the analogue incidence rule of the tangent graph. Such graph is isomorphic to $NO^{+}(6,2)$.