On the indices of convergence of reducible boolean matrices with the generalized period f0 ⩾ 2

AbstractLet f0 be the greatest common divisor of the distinct lengths of the elementary cycles of the associated digraph D(A) for a reducible boolean matrix A (f0 is called the generalized period of A). First we give a proof of the sharp bound for the indices of convergence of n × n reducible boolean matrices with the generalized period f0 ⩾ 2. Then we characterize the matrices with the largest index. The proofs are carried out by means of directed graphs