On a graph isomorphic to $N{O}^{+}(6,2)$

Abstract

Let $Q^{+}(2n-1,2)$ be a non-degenerate hyperbolic quadric of $PG(2n-1,2)$. Let $N{O}^{+}(2n,2)$ 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 $N{O}^{+}(2n-1,q)$ is a strongly regular graph. Let ${\mathcal{V}}_2^4$ be the \textit{Veronese surface} in $PG(5,q)$, and ${\mathcal{M}}_4^3$ its \textit{secant variety}. When $q=2$, $|Q^{+}(5,2)|=|{\mathcal{M}}_4^3|=35$. In this paper we define the graph $N{\mathcal{M}}_4^3$, with 28 vertices in $PG(5,2)\setminus {\mathcal{M}}_4^3$ and with the analogue incidence rule of the tangent graph. Such graph is isomorphic to $N{O}^{+}(6,2)$.

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