 
Summary: Traceminimal graphs and Doptimal weighing designs
Bernardo M. ´Abrego
Silvia Fern´andezMerchant
Michael G. Neubauer
William Watkins
California State University, Northridge
March 23, 2004, v.115
Abstract
Let G(v, ) be the set of all regular graphs on v vertices. Certain graphs from among those
in G(v, ) with maximum girth have a special property called traceminimality. In particular, all
strongly regular graphs with no triangles and some cages are traceminimal. These graphs play an
important role in the statistical theory of Doptimal weighing designs.
Each weighing design can be associated with a (0, 1)matrix. Let Mm,n(0, 1) denote the set of all
m × n (0,1)matrices and let
G(m, n) = max{det XT
X : X Mm,n(0, 1)}.
A matrix X Mm,n(0, 1) is a Doptimal design matrix if det XT
X = G(m, n). In this paper we
exhibit some new formulas for G(m, n) where n 1 (mod 4) and m is sufficiently large. These
formulas depend on the congruence class of m (mod n). More precisely, let m = nt + r where
