The Gray graph is the Levi graph of this configuration; it has a vertex for every point and every line of the configuration, and an edge for every pair of a point and a line that touch each other.

In (Monson, Pisanski, Schulte, Ivic-Weiss 2007), the Gray graph appears as a different sort of Levi graph for the edges and triangular faces of a certain locally toroidal abstract regular 4-polytope.

As with other Levi graphs, it is a bipartite graph, with the vertices corresponding to points on one side of the bipartition and the vertices corresponding to lines on the other side.

Marušič and Pisanski (2000) give several alternative methods of constructing the Gray graph.

The simplest oriented surface on which the Gray graph can be embedded has genus 7 (Marušič, Pisanski & Wilson 2005).