A problem on the exponent of primitive digraphs Original Research Article

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 γ(D(A)) = min{k set membership, variant Z+ : Ak much greater-than 0}, where Z+ denotes the set of positive integers. In a recent paper, we have proved the conjecture γ(D(A)) less-than-or-equals, slant (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 γ(D(A)) less-than-or-equals, slant (m − 1)2 + 1.