Summary: Maximum volume simplices
and trace-minimal graphs
January 11, 2004
Let (v, ) be the set of all -regular graphs on v vertices. A graph G (v, )
is trace-minimal in (v, ) 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, ). We con-
sider 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.
Keywords: volumes of simplices, regular graph, girth, generalized polygons.
1. The problem
We consider the problem of maximizing the n-dimensional volume of a sim-
plex S consisting of n + 1 vertices of the unit hypercube in m
. Without loss of
generality we assume that the origin is a vertex of S.
Let Mm,n(0, 1) be the set of all m × n matrices all of whose entries are either
0 or 1. If X is the matrix in Mm,n(0, 1) whose columns are the coordinates of the