An improvement of the Dulmage-Mendelsohn theorem

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An n × n nonnegative matrix A is called primitive if for some positive integer k, every entry in the matrix Ak is positive or, in notation, Ak ≫ 0. The exponent of primitivity of A is defined to be γ(A) = min{k ∈ ℤ+: Ak ≫ 0}, where ℤ+ denotes the set of positive integers. The well known Dulmage-Mendelsohn theorem is that γ(A) ≤ n + s(n - 2), where s is the shortest circuit in D(A), the directed graph of A. In this paper we prove that γ(A) ≤ D + 1 + s(D - 1), where D is the diameter of D(A).




Shen, J. (1996). An improvement of the Dulmage-Mendelsohn theorem. Discrete Mathematics, 158(1–3), 295–297.

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