Über die Schnittzahlenmatrix von Blockplänen. (On the intersection number matrix of block designs) (Q1119646)

Let \(B_ 1,...,B_ b\) be the blocks of a finite design (incidence structure) \({\mathbb{D}}\) with v points. Denote \(M=(\mu_{ij})\), \(\mu_{ij}=[B_ i,B_ j]\), \(Y=(k_ ik_ j)\), \(k_ i=\mu_{ii}=[B_ i]\) \((i,j=1,...,b)\). We show: \({\mathbb{D}}\) is a regular pairwise balanced design, if and only if \(rank_{{\mathbb{Q}}}M=v\geq 2\) and \(M^ 2=(r-\lambda)M+\lambda Y\) for some r,\(\lambda\in {\mathbb{Q}}\). As an application we derive Majumdar's lower and upper bounds for the intersection numbers of 2-designs. Further, we show that any blockwise semiregular group divisible design with \(b=v+c-1\), c the number of groups, is a 2-design. We also show for any 2-design the bound \(\mu_ 0\leq \lambda v/b\), \(\mu_ 0\) the smallest intersection number. This last result uses a theorem of Gerschgorin concerning the eigenvalues of complex matrices.