A bound on the exponent of primitivity in terms of diameter Original Research Article

An n × n nonnegative matrix A is called primitive if for some positive integer k, every entry in Ak is positive (Ak much greater-than 0). The exponent of primitivity of A is defined to be γ(A) = min{k set membership, variant Z+ : Ak much greater-than 0}, where Z+ denotes the set of positive integers. Two conjectures due to Hartwig, Neumann, and Lin in 1984 or so are that γ(A) less-than-or-equals, slant (m − 1)2 + 1 and γ(A) less-than-or-equals, slant D2 + 1, where m is the degree of the minimal polynomial of A and D is the diameter of the directed graph of A. It is well known that the latter is stronger than the former because D less-than-or-equals, slant m − 1. In a recent paper we have proved γ(A) less-than-or-equals, slant (m − 1)2 + 1; in this paper we prove the conjecture γ(A) less-than-or-equals, slant D2 + 1 on that basis.