On the exponent of e-regular primitive matrices

Let Pnr be the set of n-by-n r-regular primitive (0, 1)-matrices. In this paper, an explicit formula is found in terms of n and r for the minimum exponent achieved by matrices in Pnr. Moreover, matrices achieving that exponent are given in this paper. Gregory and Shen conjectured that bnr = (n/r)2 + 1 is an upper bound for the exponent of matrices in Pnr. Matrices achieving the exponent bnr are presented for the case when n is not a multiple of r. In particular, it is shown that b2r+1,r is the maximum exponent attained by matrices in P2r+1,r. When n is a multiple of r, it is conjectured that the maximum exponent achieved by matrices in Pnr is strictly smaller than bnr. Matrices attaining the conjectured maximum exponent in that set are presented. It is shown that the conjecture is true when n = 2r.