Levi Graph for PG(2,3)

This represents the point-line incidence graph (also known as the Levi graph) of the projective plane of order 3.  Thus, there are 26 vertices, representing the points and lines, and a pair of vertices are connected by an edge exactly when the point and line that they represent in the geometry are incident.  Since every line has 4 points and 4 lines pass through every point, this is a regular graph of degree 4.  This particular realization illustrates two "layers" consisting of 13 vertices each that are connected in an elegant honeycomb pattern.