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ON THE GAPS IN THE SET OF EXPONENTS OF BOOLEAN PRIMITIVE CIRCULANT MATRICES.
 

Summary: ON THE GAPS IN THE SET OF EXPONENTS OF BOOLEAN PRIMITIVE
CIRCULANT MATRICES.
M.I. BUENO AND S. FURTADO
Abstract. In this paper we consider the problem of describing the possible exponents of boolean primitive
circulant matrices. We give a conjecture for the possible such exponents and prove this conjecture in several cases.
In particular, we consider in greater detail the case of matrices whose generating vector has three nonzero entries.
Key words. Circulant primitive matrices, exponent of primitive matrices, circulant digraphs, basis for a group.
AMS subject classifications. 11P70, 05C25, 05C50.
1. Introduction. A Boolean matrix is a matrix over the binary Boolean algebra {0, 1}.
An n-by-n Boolean matrix C is said to be circulant if each row of C (except the first one) is
obtained from the preceding row by shifting the elements cyclically 1 column to the right. In
other words, the entries of a circulant matrix C = (cij) are related in the manner: ci+1,j =
ci,j-1, where 0 i n - 2, 0 j n - 1, and the subscripts are computed modulo n.
The first row of C is called the generating vector. Here and throughout we number the rows
and columns of an n-by-n matrix from 0 to n - 1.
The set of all n-by-n Boolean circulant matrices forms a multiplicative commutative
semigroup Cn with |Cn| = 2n
[3, 8]. In 1974, K. H. Kim-Buttler and J.R. Krabill [6], and S.
Schwarz [9] investigated this semigroup thoroughly.
An n-by-n Boolean matrix C is said to be primitive if there exists a positive integer k

  

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

 

Collections: Mathematics