# On the exponent of R-regular primitive matrices

## Main Article Content

## Abstract

Let P_{nr} 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 P_{nr}. Moreover, matrices achieving that exponent are given in this paper. Gregory and Shen conjectured that b_{nr} = (*n*/*r)*^{2} + 1 is an upper bound for the exponent of matrices in P_{nr}. Matrices achieving the exponent bnr are presented for the case when *n* is not a multiple of *r*. In particular, it is shown that *b _{2r}*

_{+1,r}is the maximum exponent attained by matrices in

*P*. When

_{2r}_{+1,r}*n*is a multiple of

*r*, it is conjectured that the maximum exponent achieved by matrices in

*P*is strictly smaller than

_{nr}*b*. Matrices attaining the conjectured maximum exponent in that set are presented. It is shown that the conjecture is true when

_{nr}*n*=

*2r*.