Lehman matrices

Authors:
Gérard Cornuéjols;Bertrand Guenin;Levent Tunçel
Affiliations:
Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213, USA and LIF, Faculté des Sciences de Luminy, Université d' Aix-Marseille, 13288 Marseille, France;Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada;Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Venue:
Journal of Combinatorial Theory Series B
Year:
2009

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Abstract

A pair of square 0,1 matrices A,B such that AB^T=E+kI (where E is the nxn matrix of all 1s and k is a positive integer) are called Lehman matrices. These matrices figure prominently in Lehman's seminal theorem on minimally nonideal matrices. There are two choices of k for which this matrix equation is known to have infinite families of solutions. When n=k^2+k+1 and A=B, we get point-line incidence matrices of finite projective planes, which have been widely studied in the literature. The other case occurs when k=1 and n is arbitrary, but very little is known in this case. This paper studies this class of Lehman matrices and classifies them according to their similarity to circulant matrices.