The trace-minimal graph with 2v2v vertices and regularity v+1v+1

In this paper we find the trace-minimal graph in the class of (v+1)(v+1)-regular graphs with 2v2v vertices. It is proved that the join of an empty graph on v-1v-1 vertices and a cycle graph on v+1v+1 vertices is the unique trace-minimal graph in this class. These findings have applications to experimental designs in statistics, and to the geometry of the unit hypercube in higher dimensions. We present these applications.