Maximum volume simplices and trace-minimal graphs

Let (v, δ) be the set of all δ-regular graphs on v vertices. A graphG ∈ (v, δ) is trace-minimal in (v, δ) if the vector whose ith entry is the trace of the ith power of the adjacency matrix of G, is minimum under the lexicographic order among all such vectors corresponding to graphs in (v, δ). We consider the problem of maximizing the volume of an n-dimensional simplex consisting of n + 1 vertices of the unit hypercube in m. We show that if n ≡ −1 (mod 4), such a maximum can be explicitly evaluated for all m large enough whenever an appropriate trace-minimal graph is known.