On Orthogonal Graphs and their Automorphisms

We provide a characterization of the connected subgraphs of the graphs with vertex set the non-isotropic points in a quadratic space $(V,Q)$, two points adjacent if and only if they span a tangent line.

Here $(V,Q)$ is a quadratic space $V$ over a finite field $\mathbb{F}_q$ of order $q$, where $q>3$ is odd, equipped with a non-degenerate quadratic form $Q$.

The local structure of the graph (i.e. the graph induced on the neighbors of a point) determines the structure of the full graph.

This characterization helps to determine the automorphism group of these graphs.