Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415982 | Linear Algebra and its Applications | 2016 | 13 Pages |
Let B(n,k) be the set of all (0,1)-matrices of order n with constant line sum k and let νË(n,k) be the minimum rank over B(n,k). It is known that ân/kââ¤Î½Ë(n,k)â¤Î½Ë(n,k)â¤ân/kâ+k, where νË(n,k) is the rank of a recursively defined matrix AËâB(n,k). Brualdi, Manber and Ross showed that νË(n,k)=ân/kâ if and only if k|n. In this paper, we show that νË(n,k)=ân/kâ+k if and only if (n,k) satisfies one of the following three relations: (i) nâ¡Â±1(modk), k=2 or 3; (ii) n=k+1, kâ¥2; (iii) n=4q+3, k=4 and qâ¥1. Moreover, we obtain the exact values of νË(n,4) for all nâ¥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 νË(n,k)<νË(n,k)<ân/kâ+k, which gives a positive answer to a question posed by Pullman and Stanford.