 
Summary: EXPONENTS OF PRIMITIVE CIRCULANT MATRICES
A boolean matrix is a matrix whose entries are zeros and ones and for
which the addition and multiplication are defined as follows: 0 + 0 = 0,
0 + 1 = 1 + 0 = 1, 1 + 1 = 1, 0 0 = 0, 0 1 = 0, and 1 1 = 1. An nbyn
boolean matrix A is said to be circulant if each row of A (except the first) is
obtained from the preceding row by shifting the elements cyclically 1 column
to the right.
A nbyn boolean circulant matrix C is said to be primitive if there exists
a positive integer k such that Ck
= J, where J is the nbyn matrix whose
entries are all ones and the product is computed in the algebra {0, 1}. The
smallest such k is called the exponent of C.
An interesting open problem is to determine all possible exponents at
tained by nbyn boolean primitive circulant matrices.
In the literature, this problem has been considered. However, not much
progress has been done. It has been proven that shown if C is an nbyn
boolean circulant primitive matrix, then its exponent is either n  1, n/2 ,
n/2  1 or does not exceed n/3 + 1. The matrices with exponents n  1,
n/2 , n/2  1 are also characterized.
Our problem can also be stated in terms of Cayley digraphs: A boolean
