EXPONENTS OF PRIMITIVE CIRCULANT MATRICES A boolean matrix is a matrix whose entries are zeros and ones and for 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 n-by-n 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 n-by-n boolean circulant matrix C is said to be primitive if there exists a positive integer k such that Ck = J, where J is the n-by-n 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 n-by-n 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 n-by-n 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 Collections: Mathematics