A problem on the exponent of primitive digraphs

A digraph D(A) is called primitive if and only if A, the (0, 1) connection matrix of D(A), is primitive. The exponent of primitivity of D(A) is defined to be gamma(D(A)) = min{k is an element of Z(+): A(k) much greater than 0}, where Z(+) denotes the set of positive integers. In a recent paper, we have proved the conjecture gamma(D(A)) less than or equal to (m - 1)(2) + 1 due to Robert E. Hartwig and Michael Neumann, where m is the degree of the minimal polynomial of A. In this paper, we characterize the equality case of the upper bound gamma(D(A)) less than or equal to (m - 1)(2) + 1.