Geometric matrices and an inequality for (0,1)-matrices

Abstract

Let A be an m × n (0, 1)-matrix with row vectors R 1,…, R m and column vectors C 1,…, C n . If there exist integers α, β such that R i R j = α whenever R i ≠ R j and C i C j = β whenever C i ≠ C j , then A will be called geometric. ( R iR j , C iC j are the usual dot products of the vectors involved.) The geometric matrices are classified, and it is shown that (apart from certain trivialities) every geometric matrix is based on a symmetric balanced incomplete block design. Assume that each column of A has a zero entry and that C i ≠ C j for some i and j. Under these assumptions it is shown that m · min{R iR j:R i≠R j}⩽n · max{C iC j:C i≠C j} , and that equality occurs if and only if A is geometric. The results generalize a theorem of de Bruijn and Erdös concerning combinatorial designs.