Further results on the minimum rank of regular classes of (0,1)-matrices

Let B(n,k) be the set of all (0,1)-matrices of order n with constant line sum k and let ν˜(n,k) be the minimum rank over B(n,k). It is known that ⌈n/k⌉≤ν˜(n,k)≤νˆ(n,k)≤⌊n/k⌋+k, where νˆ(n,k) is the rank of a recursively defined matrix Aˆ∈B(n,k). Brualdi, Manber and Ross showed that ν˜(n,k)=⌈n/k⌉ if and only if k|n. In this paper, we show that ν˜(n,k)=⌊n/k⌋+k if and only if (n,k) satisfies one of the following three relations: (i) n≡±1(modk), k=2 or 3; (ii) n=k+1, k≥2; (iii) n=4q+3, k=4 and q≥1. Moreover, we obtain the exact values of ν˜(n,4) for all n≥4 and determine all the possible ranks of regular (0,1)-matrices in B(n,4). We also present some positive integer pairs (n,k) such that ν˜(n,k)<νˆ(n,k)<⌊n/k⌋+k, which gives a positive answer to a question posed by Pullman and Stanford.