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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

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics