Summary: ON THE EXPONENT OF R-REGULAR PRIMITIVE MATRICES.
M.I. BUENO AND S. FURTADO
Abstract. Let Pnr be the set of n-by-n r-regular primitive (0, 1)-matrices. In this paper we
find an explicit formula in terms of n and r for the minimum exponent achieved by matrices in
Pnr. Moreover, we give matrices achieving that exponent. Gregory and Shen  conjectured that
bnr = n
2 +1 is an upper bound for the exponent of matrices in Pnr. We present matrices achieving
the exponent bnr when n is not a multiple of r. In particular, we show that b2r+1,r is the maximum
exponent attained by matrices in P2r+1,r. When n is a multiple of r we conjecture that the maximum
exponent achieved by matrices in Pnr is strictly smaller than bnr and give matrices attaining the
conjectured maximum exponent in that set. We also show that our conjecture is true when n = 2r.
Key words. r-regular matrices, primitive matrices, exponent of primitive matrices.
AMS subject classifications. 05C20, 05C50, 15A36.
1. Introduction. A nonnegative square matrix A is called primitive if there
exists a positive integer k such that Ak
is positive. The smallest such k is called the
exponent of A. We denote the exponent of a primitive matrix A by exp(A).
A (0,1)-matrix A is said to be r-regular if every column sum and every row sum
is constantly r.