Degree of indecomposability of certain highly regular zero-one matrices Original Research Article

A v × v (0, 1) matrix M is said to be r-indecomposable if every i × j submatrix of zeros satisfies i + j less-than-or-equals, slant v − r. This concept is a natural generalization of Hall matrices (0-indecomposable) and fully indecomposable matrices (1-indecomposable). Letting δ(M) denote the largest r such that M is r-indecomposable, our aim is to compute δ(M) for matrices having constant row and column sums k. In this case it is known that 0 less-than-or-equals, slant δ(M) less-than-or-equals, slant k − 1. Our principal result is that certain “highly regular” matrices (incidence matrices of symmetric designs and partial λ-geometries) must have δ(M) = k − 1. This leads us to conjecture that the same conclusion holds for the more general class of (incidence matrices of) bipartite distance-regular graphs.