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Trace-minimal graphs and D-optimal weighing designs Bernardo M. Abrego
 

Summary: Trace-minimal graphs and D-optimal weighing designs
Bernardo M. ´Abrego
Silvia Fern´andez-Merchant
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 trace-minimality. In particular, all
strongly regular graphs with no triangles and some cages are trace-minimal. These graphs play an
important role in the statistical theory of D-optimal 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 D-optimal 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

  

Source: Abrego, Bernardo - Department of Mathematics, California State University, Northridge
Fernandez, Silvia - Department of Mathematics, California State University, Northridge

 

Collections: Mathematics