On circulant Boolean matrices

Abstract

Let C be an n × n circulant Boolean matrix. If C is primitive, we prove that its exponent γ( C) = n − 1 or γ(C) ⩽ [ n 2 ] and list all C with γ( C) = n − 1 or [ n 2 ] when n > 9. In the general case, we express the period of C explicitly and show that the index of convergence of C equals the exponent of a related primitive circulant Boolean matrix and that the index of maximum density of C equals the index of convergence of C.